# Multiresolution Hilbert Approach to Multidimensional Gauss-Markov Processes

Authors: Thibaud Taillefumier; Jonathan Touboul
Journal: International Journal of Stochastic Analysis
 (2011)

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## Abstract

The study of the multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined through the theory of stochastic integration. Here, inspired by the Lévy-Ciesielski construction of the Wiener process, we propose an alternative representation of multidimensional Gauss-Markov processes as expansions on well-chosen Schauder bases, with independent random coefficients of normal law with zero mean and unit variance. We thereby offer a natural multiresolution description of the Gauss-Markov processes as limits of finite-dimensional partial sums of the expansion, that are strongly almost-surely convergent. Moreover, such finite-dimensional random processes constitute an optimal approximation of the process, in the sense of minimizing the associated Dirichlet energy under interpolating constraints. This approach allows for a simpler treatment of problems in many applied and theoretical fields, and we provide a short overview of applications we are currently developing.


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## Body

## 1. Introduction

Intuitively, multidimensional continuous stochastic processes are easily conceived as solutions to randomly perturbed differential equations of the form(1.1)Ẋt=f(Xt,t,ξt),
where the perturbative term ξt implicitly defines a probability space and f satisfies some ad hoc regularity conditions. If the existence of such processes is well established for a wide range of equations through the standard Itô integration theory (see, e.g., [1]), studying their properties proves surprisingly challenging, even for the simplest multidimensional processes. Indeed, the high dimensionality of the ambient space and the nowhere differentiability of the sample paths conspire to heighten the intricacy of the sample paths spaces. In this regard, such spaces have been chiefly studied for multidimensional diffusion processes [2], and more recently, the development of rough paths theory has attracted renewed interest in the field (see [3–7]). However, aside from these remarkable theoretical works, little emphasis is put on the sample paths since most of the available results only make sense in distribution. This is particularly true in the Itô integration theory, where the sample path is completely neglected for the Itô map being defined up to null sets of paths.To overcome the difficulty of working in complex multidimensional spaces, it would be advantageous to have a discrete construction of a continuous stochastic process as finite-dimensional distributions. Since we put emphasis on the description of the sample paths space, at stake is to write a processX as an almost surely pathwise convergent series of random functions(1.2)Xt=lim⁡N→∞XtNwithXtN=∑n=0Nfn(t)⋅Ξn,
where fn is a deterministic function and Ξn is a given random variable.The Lévy-Ciesielski construction of thed-dimensional Brownian motion W (also referred to as Wiener process) provides us with an example of discrete representation for a continuous stochastic process. Noticing the simple form of the probability density of a Brownian bridge, it is based on completing sample paths by interpolation according to the conditional probabilities of the Wiener process [8]. More specifically, the coefficients Ξn are Gaussian independent and the elements fn, called the Schauder elements and denoted by sn, are obtained by time-dependent integration of the Haar basis elements: s0,0(t)=tId and sn,k(t)=sn,k(t)Id, with for all n>0(1.3)sn,k(t)={2(n-1)/2(t-ln,k),k2-n+1≤t≤(2k+1)2-n,2(n-1)/2(rn,k-t),(2k+1)2-n≤t≤(k+1)2-n+1,0,otherwise.
This latter point is of relevance since, for being a Hilbert system, the introduction of the Haar basis greatly simplifies the demonstration of the existence of the Wiener process [9]. From another perspective, fundamental among discrete representations is the Karhunen-Loève decomposition giving a representation of stochastic processes by expanding it on a basis of orthogonal functions [10, 11]. The definition of the basis elements fn depends only on the second-order statistics of the considered process and the coefficients ξn are pairwise uncorrelated random variables. Incidentally, such a decomposition is especially suited to study the Gaussian processes because the coefficients of the representation are Gaussian and independent. For these reasons, the Karhunen-Loève decomposition is of primary importance in exploratory data analysis, leading to methods referred to as “principal component analysis,” “Hotelling transform" [12] or “proper orthogonal decomposition” [13] according to the field of application. In particular, it was directly applied to the study of the stationary Gaussian Markov processes in the theory of random noise in radio receivers [14].It is also important for our purpose to realize that the Schauder elementssn have compact supports that exhibit  a nested structure: this fact entails that the finite sums WN are processes that interpolate  the limit process W on the endpoints of the supports, that is, on the dyadic points k2-N, 0≤k≤2N. One of the specific goal of our construction is to maintain such a property in the construction of all multidimensional  the Gauss-Markov processes X (i.e., processes that are both Gaussian and satisfy the Markov property) of the form:(1.4)Xt=g(t)∫0tf(s)⋅dWs
(covering all 1-dimensional Gauss-Markov processes thanks to Doob’s representation of Gauss-Markov processes), being successively approximated by finite-dimensional processes XN that interpolates X at ever finer resolution. In that respect, it is only in that sense that we refer to our framework as a multiresolution approach as opposed to the wavelet multiresolution theory [15]. Other multiresolution approaches have been developed for certain Gaussian processes, most notably for the fractional Brownian motion [16].In view of this, we propose a construction of the multidimensional Gaussian Markov processes using a multiresolution Schauder basis of functions. As for the Lévy-Ciesielski construction, and in contrast with Karhunen-Loève decomposition, our basis is not made of orthogonal functions but the elements are of nested compact support and the random coefficientsΞn are always independent and Gaussian (for convenience with law 𝒩(0,Id), i.e., with zero mean and unitary variance). We first develop a heuristic approach for the construction of stochastic processes reminiscent of the midpoint displacement technique [8, 9], before rigorously deriving the multiresolution basis that we will be using   the paper. This set of functions is then studied as a multiresolution Schauder basis of functions: in particular, we derive explicitly from the multiresolution basis an Haar-like Hilbert basis, which is the underlying structure explaining the dual relationship between basis elements and coefficients. Based on these results, we study the construction application and its inverse, the coefficient applications, that relate coefficients on the Schauder basis to sample paths. We follow up by proving the almost-sure and strong convergence of the process having independent standard normal coefficients on the Schauder basis to a Gauss-Markov process. We also show that our decomposition is optimal in some sense that is strongly evocative of spline interpolation theory [17]: the construction yields successive interpolations of the process at the interval endpoints that minimize  the Dirichlet energy induced by the differential operator associated with the Gauss-Markov process [18, 19]. We also provide a series of examples for which the proposed Schauder framework yields bases of functions that have simple closed form formulae: in addition to the simple one-dimensional Markov processes, we explicit our framework for two classes of multidimensional processes, the Gauss-Markov rotations and the iteratively integrated Wiener processes (see, e.g., [20–22]).The ideas underlying this work can be directly traced back to the original work of Lévy. Here, we intend to develop a self-contained Schauder dual framework to further the description of multidimensional Gauss-Markov processes, and, in doing so, we extend some well-known results of interpolation theory in signal processing [23–25]. To our knowledge, such an approach is yet to be proposed. By restraining our attention to the Gauss-Markov processes, we obviously do not assume generality. However, we hope our construction proves of interest for a number of points, which we tentatively list in the following. First, the almost-sure pathwise convergence of our construction together with the interpolation property of the finite sums allows to reformulate results of the stochastic integration in term of the geometry of finite-dimensional sample paths. In this regard, we found it appropriate to illustrate how in our framework, the Girsanov theorem for the Gauss-Markov processes appears as a direct consequence of the finite-dimensional change of variable formula. Second, the characterization of our Schauder elements as the minimizer of a Dirichlet form paves the way to the construction of infinite-dimensional Gauss-Markov processes, that is, processes whose sample points themselves are infinite-dimensional [26, 27]. Third, our construction shows that approximating a Gaussian process by a sequence of interpolating processes relies entirely on the existence of a regular triangularization of the covariance operator, suggesting to further investigate this property for non-Markov Gaussian processes [28]. Finally, there is a number of practical applications where applying the Schauder basis framework clearly provides an advantage compared to standard stochastic calculus methods, among which first-hitting times of stochastic processes, pricing of multidimensional path-dependant options [29–32], regularization technique for support vector machine learning [33], and more theoretical work on uncovering the differential geometry structure of the space of the Gauss-Markov stochastic processes [34]. We conclude our exposition by developing in more detail  some of these direct implications which will be the subjects of forthcoming papers.


## 2. Heuristic Approach to the Construction

In order to provide a discrete multiresolution description of the Gauss-Markov processes, we first establish basic results about the law of the Gauss-Markov bridges in the multidimensional setting. We then use them to infer the candidate expressions for our desired bases of functions, while imposing its elements to be compactly supported on   nested sequence segments. Throughout this paper, we are working in a complete probability space(Ω,ℱ,P).


### 2.1. Multidimensional Gauss-Markov Processes

After recalling the definition of the multidimensional Gauss-Markov processes in terms of stochastic integral, we use the well-known conditioning formula for the Gaussian vectors to characterize the law of the Gauss-Markov bridge processes.


#### 2.1.1. Notations and Definitions

Let(Wt,ℱt,t∈[0,1]) be an m-dimensional Wiener process, consider the continuous functions α:[0,1]→ℝd×dΓ:[0,1]→ℝd×m, and define the positive bounded continuous function Γ=Γ·ΓT:[0,1]→ℝd×d. The d-dimensional Ornstein-Uhlenbeck process associated with these parameters is solution of the equation(2.1)dXt=α(t)⋅Xtdt+Γ(t)⋅dWt,
and with initial condition Xt0 in t0, it reads(2.2)Xt=F(t0,t)⋅Xt0+F(t0,t)⋅∫t0tF(s,t0)⋅Γ(s)⋅dWs,
where F(t0,t) is the flow of the equation, namely, the solution in ℝd×d of the linear equation:(2.3)∂F(t0,t)∂t=α(t)F(t),F(t0,t0)=Id.
Note that the flow F(t0,t) enjoys the chain rule property:(2.4)F(t0,t)=F(t1,t)⋅F(t0,t1).
For all t,s such that t0<s,t, the vectors Xt and Xs admit the covariance(2.5)Ct0(s,t)=F(t0,t)(∫t0t∧sF(w,t0)Γ(w)F(w,t0)Tdw)F(t0,s)T=F(t0,t)ht0(s,t)F(t0,s)T,
where we further defined hu(s,t) the function(2.6)hu(s,t)=∫stF(w,u)⋅Γ(w)⋅F(w,u)Tdw,
which will be of particular interest in the sequel. Note that because of the chain rule property of the flow, we have (2.7)hv(s,t)=F(v,u)hu(s,t)F(v,u)T.
We suppose that the process X is never degenerated, that is, for all t0<u<v, all the components of the vector Xv taking into account Xu are nondeterministic random variables, which is equivalent to saying that the covariance matrix of Xv taking into account Xu, denoted by Cu(v,v) is symmetric positive definite for any u≠v. Therefore, assuming the initial condition X0=0, the multidimensional centered process X has a representation (similar to Doob’s representation for one-dimensional processes, see [35]) of form(2.8)Xt=g(t)∫0tf(s)⋅dWs,
with g(t)=F(0,t) and f(t)=F(t,0)·Γ(t).Note that the processes considered in this paper are defined on the time interval[0,1]. However, because of the time-rescaling property of these processes, considering the processes on this time interval is equivalent to considering the process on any other bounded interval without loss of generality.


#### 2.1.2. Conditional Law and Gauss-Markov Bridges

As stated in the introduction, we aim at defining a multiresolution description of Gauss-Markov processes. Such a description can be seen as a multiresolution interpolation of the process that is getting increasingly finer. This principle, in addition to the Markov property, prescribes to characterize the law of the corresponding Gauss-Markov bridge, that is, the Gauss-Markov process under consideration, conditioned on its initial and final values. The bridge process of the Gauss process is still a Gauss process and, for a Markov process, its law can be computed as follows.Proposition 2.1.
Lettx≤tz two times in the interval [0,1]. For any t∈[tx,tz], the random variable Xt conditioned on Xtx=x and Xtz=z is a Gaussian variable with covariance matrix Σ(t) and mean vector μ(t) given by
(2.9)𝚺(t;tx,tz)=ht(tx,t)(ht(tx,tz))-1ht(t,tz),μ(t)=μl(t;tx,tz)⋅x+μr(t;tx,tz)⋅z,
where the continuous matrix functions μl(·;tx,tz) and μr(·;tx,tz) of ℝd×d are given by
(2.10)μl(t;tx,tz)=F(tx,t)htx(t,tz)(htx(tx,tz))-1,μr(t;tx,tz)=F(tz,t)htz(tx,t)(htz(tx,tz))-1.Note that the functionsμl and μr have the property that μl(tx;tx,tz)=μr(tz;tx,tz)=Id and μl(tz;tx,tz)=μr(tx;tx,tz)=0 ensuring that the process is indeed equal to x at time tx and z at time tz.Proof.
Lettx,tz be two times of the interval [0,1] such that tx<tz, and let t∈[tx,tz]. We consider the Gaussian random variable ξ=(Xt,Xtz) conditioned on the fact that Xtx=x. Its mean can be easily computed from   expression (2.2) and reads
(2.11)(mt,mtz)=(F(tx,t)x,F(tx,tz)x)=(g(t)g-1(tx)x,g(tz)g-1(tx)x),
and its covariance matrix, from (2.5), reads
(2.12)[Ct,tCt,tzCtz,tCtz,tz]=[F(tx,t)htx(tx,t)F(tx,t)TF(tx,t)htx(tx,t)F(tx,tz)TF(tx,tz)htx(tx,t)F(tx,t)TF(tx,tz)htx(tx,tz)F(tx,tz)T]=[ht(tx,t)ht(tx,t)F(t,tz)TF(t,tz)ht(tx,t)F(t,tz)ht(tx,tz)F(t,tz)T].
From there, we apply the conditioning formula for the Gaussian vectors (see, e.g., [36]) to infer the law of Xt conditioned on Xtx=x and Xtz=z, that is the law 𝒩(μ(t),Σ(t;tx,tz)) of Bt where B denotes the bridge process obtained by pinning X in tx and tz. The covariance matrix is given by
(2.13)𝚺(t;tx,tz)=Cy,y-Cy,zCz,z-1Cz,y=ht(tx,t)-ht(tx,t)(ht(tx,tz))-1ht(tx,t)=ht(tx,t)(ht(tx,tz))-1ht(t,tz),
and the mean reads
(2.14)μ(t)=my+Cy,zCz,z-1(z-mz)=F(tx,t)(Id-htx(tx,t)(htx(tx,tz))-1)x+F(tz,t)htz(tx,t)(htz(tx,tz))-1z,=F(tx,t)htx(t,tz)(htx(tx,tz))-1︸μl(t;tx,tz)⋅x+F(tz,t)htz(tx,t)(htz(tx,tz))-1︸μr(t;tx,tz)⋅z,
where we have used the fact that htx(tx,tz)=htz(tx,t)+htx(t,tz). The regularity of the thus-defined functions μx and μz directly stems from the regularity of the flow operator F. Moreover, since for any 0≤t,u≤1, we observe that F(t,t)=Id and hu(t,t)=0; we clearly have μx(tx)=μy(t)=Id and μx(t)=μy(tx)=0.Remark 2.2.
Note that these laws can also be computed using the expression of the density of the processes  but involve  more intricate calculations. An alternative approach also provides a representation of Gauss-Markov bridges with the use of integral and anticipative representation [37]. These approaches allow to compute the probability distribution of the Gauss-Markov bridge as a process (i.e., allows to compute the covariances), but since this will be of no use in the sequel, we do not provide the expressions.


### 2.2. The Multiresolution Description of Gauss-Markov Processes

Recognizing the Gauss property and the Markov property as the two crucial elements for a stochastic process to be expanded to Lévy-Cesielski, our approach first proposes to exhibit bases of deterministic functions that would play the role of the Schauder bases for the Wiener process. In this regard, we first expect such functions to be continuous and compactly supported on increasingly finer supports (i.e., subintervals of the definition interval[0,1]) in a similar nested binary tree structure. Then, as in the Lévy-Ciesielski construction, we envision that, at each resolution (i.e., on each support), the partially constructed process (up to the resolution of the support) has the same conditional expectation as the Gauss-Markov process when conditioned on the endpoints of the supports. The partial sums obtained with independent Gaussian coefficients of law 𝒩(0,1) will thus approximate the targeted Gauss-Markov process in a multiresolution fashion, in the sense that, at every resolution, considering these two processes on the interval  endpoints yields finite-dimensional Gaussian vectors of the same law.


#### 2.2.1. Nested Structure of the Sequence of Supports

Here, we define the nested sequence of segments that constitute the supports of the multiresolution basis. We construct such a sequence by recursively partitioning the interval[0,1].More precisely, starting fromS1,0=[l1,0,r1,0] with l1,0=0 and r1,0=1, we iteratively apply the following operation. Suppose that, at the nth step, the interval [0,1] is decomposed into 2n-1 intervals Sn,k=[ln,k,rn,k], called supports, such that ln,k+1=rn,k for 0≤k<2n-1. Each of these intervals is then subdivided into two child intervals, a left-child Sn+1,2k and a right-child Sn+1,2k+1, and the subdivision point rn+1,2k=ln+1,2k+1 is denoted by mn,k. Therefore, we have defined three sequences of real ln,k, mn,k, and rn,k for n>0 and 0≤k<2n-1 satisfying l0,0=0≤ln,k<mn,k<rn,k≤r0,0=1 and(2.15)ln+1,2k=ln,k,mn,k=rn+1,2k=ln+1,2k+1,rn+1,2k+1=rn,k
with the convention   l0,0=0 and r0,0=1 and S0,0=[0,1]. The resulting sequence of supports {Sn,k;n≤0,0≤k<2n-1} clearly has a binary tree structure.For the sake of compactness of notations, we defineℐ the set of indices(2.16)I=⋃n<NInwithIN={(n,k)∈N2∣0<n≤N,0≤k<2n-1},
and for N>0, we define DN={mn,k,(n,k)∈ℐN-1}∪{0,1}, the set of endpoints of the intervals SN,k. We additionally require that there exists ρ∈(0,1) such that for all (n,k)∈ℐmax(rn,k-mn,k,mn,k-ln,k)<ρ(rn,k-ln,k) which in particular implies that(2.17)lim⁡n→∞sup⁡krn,k-ln,k=0
and ensures that the set of endpoints ∪N∈NDN is everywhere dense in [0,1]. The simplest case of such partitions is the dyadic partition of [0,1], where the endpoints for (n,k)∈ℐ read(2.18)ln,k=k2-n+1,mn,k=(2k+1)2-n,rn,k=(k+1)2-n+1,
in which case the endpoints are simply the dyadic points ∪NDN={k2-N|0≤k≤2N}.  Figure 1 represents the global architecture of the nested sequence of intervals.Figure 1
A sequence of nested intervals.The nested structure of the supports, together with the constraint of continuity of the bases elements, implies that only a finite number of coefficients are needed to construct the exact value of the process at a given endpoint, thus providing us with an exact schema to simulate the sample values of the process on the endpoint up to an arbitrary resolution, as we will further explore.


#### 2.2.2. Innovation Processes for Gauss-Markov Processes

ForXt, a multidimensional Gauss-Markov process, we call the multiresolution description of a process the sequence of conditional expectations on the nested sets of endpoints Dn. In detail, if we denote by ℱN the filtration generated by {Xt;t∈DN} given the values of the process at the endpoints DN of the partition, we introduce the sequence of the Gaussian processes (ZtN)N≥1 defined by:(2.19)ZtN=E[Xt∣FN]=EN[Xt].
These processes ZN can be naturally viewed as an interpolation of the process X sampled at the increasingly finer times DN  since for all t∈DN we have ZtN=XtN. The innovation process (δtN,ℱt,t∈[0,1]) is defined as the update transforming the process ZtN into ZtN+1, that is,
(2.20)δtN=ZtN+1-ZtN.
It corresponds to the difference the additional knowledge of the process at the points mN,k make on the conditional expectation of the process. This process satisfies the following important properties that found our multiresolution construction.Proposition 2.3.
The innovation processδtN is a centered Gaussian process independent of the processes Ztn for any n≤N. For s∈SN,k and t∈SN,p with k,p∈ℐN, the covariance of the innovation process reads
(2.21)EN[δtN⋅(δsN)T]={μN,k(t)⋅𝚺N,k⋅μN,k(t)Tifk=p,0ifk≠p,
where
(2.22)μN,k(t)={μr(t;lN,k,mN,k),t∈[lN,k,mN,k],μl(t;mN,k,rN,k),t∈[mN,k,rN,k].
with μl, μr and ΣN,k=Σ(mN,k;lN,k,rN,k) as defined in Proposition 2.1.Proof.
Because of the Markovian property of the processX, the law of the process ZN can be computed from the bridge formula derived in Proposition 2.1 and we have
(2.23)ZtN=μl(t;lN,k,rN,k)⋅XlN,k+μr(t;lN,k,rN,k)⋅XrN,k,ZtN+1={μl(t;lN,k,mN,k)⋅XlN,k+μr(t;lN,k,mN,k)⋅XmN,k,fort∈[lN,k,mN,k],μl(t;mN,k,rN,k)⋅XmN,k+μr(t;mN,k,rN,k)⋅XrN,k,fort∈[lN,k,mN,k].
Therefore, the innovation process can be written for t∈SN,k as
(2.24)δtN=μN,kN(t)⋅XmN,k+νN(t)⋅QtN,
where QtN is a ℱN measurable process  νN(t) a deterministic matrix function and
(2.25)μN,k(t)={μr(t;lN,k,mN,k),t∈[lN,k,mN,k],μl(t;mN,k,rN,k),t∈[mN,k,rN,k].
The expressions of ν and Q are quite complex  but are highly simplified when noting that
(2.26)E[δtN∣FN]=E[ZtN+1∣FN]-ZtN=E[E[Zt∣FN+1]∣FN]-ZtN=0
directly implying that ν(t)·QtN=μN(t)·ZmN,kN and yielding the remarkably compact expression
(2.27)δtN=μN,k(t)⋅(XmN,k-ZmN,kN).
This process is a centered Gaussian process. Moreover, observing that it is ℱN-measurable, it can be written as
(2.28)δtN=μN,k(t)⋅({XmN,k∣FN}-ZmN,kN),
and the process {XmN,k∣ℱN} appears as the Gauss-Markov bridge conditioned at times lN,k and rN,k, and whose covariance is given by Proposition 2.1 and that has the expression
(2.29)𝚺N,k=𝚺(mN,k;lN,k,rN,k)=hmn,k(ln,k,mn,k)(hmn,k(ln,k,rn,k))-1hmn,k(mn,k,rn,k).
Let (s,t)∈[0,1]2, and assume that s∈SN,k and t∈SN,p. If k≠p, then, because of the Markov property of the process X, the two bridges are independent and therefore the covariance 𝔼N[δtN·(δsN)T] is zero. If k=p, we have
(2.30)EN[δtN⋅(δsN)T]=μN,k(t)⋅𝚺N,k⋅μN,k(s)T.
Eventually, the independence property stems from the simple properties of the conditional expectation. Indeed, let n≤N. We have
(2.31)E[Ztn⋅(δsN)T]=E[Ztn⋅(ZsN+1-ZsN)T]=E[E[Xt∣Fn]⋅(E[XsT∣FN+1]-E[XsT∣FN])]=E[E[Xt∣Fn]⋅E[XsT∣FN+1]]-E[E[Xt∣Fn]⋅E[XsT∣FN]]=E[Ztn(Zsn)T]-E[Ztn(Zsn)T]=0
and the fact that a zero covariance between two Gaussian processes implies the independence of these processes concludes the proof.


#### 2.2.3. Derivation of the Candidate Multiresolution Bases of Functions

We deduce from the previous proposition the following fundamental theorem of this paper.Theorem 2.4.
For allN∈ℕ, there exists a collection of   ψN,k:[0,1]↦ℝd×d that are zero outside the subinterval SN,k   such that in distribution one has:
(2.32)δtN=∑k∈INψN,k(t)⋅ΞN,k,
where ΞN,k are independent d-dimensional standard normal random variables (i.e., of law 𝒩(0,Id)). This basis of functions is unique up to an orthogonal transformation.Proof.
The two processesδtN and dtN=def∑k∈ℐNψN,k(t)·ΞN,k are two Gaussian processes of mean zero. Therefore, we are searching for functions ψN,k vanishing outside SN,k and ensuring that the two processes have the same probability distribution. A necessary and sufficient condition for the two processes to have the same probability distribution is to have the same covariance function (see, e.g., [36]). We therefore need to show the existence of a collection of functions ψN,k(t) functions that vanish outside the subinterval SN,k and that ensure that the covariance of the process dN is equal to the covariance of δN. Let (s,t)∈[0,1] such that s∈SN,k and t∈SN,p. If k≠p, the assumption fact that the functions ψN,k vanish outside SN,k implies that
(2.33)E[dtN⋅(dsN)T]=0.
If k=p, the covariance reads
(2.34)E[dtN⋅(dsN)T]=E[ψN,k(t)⋅ΞN,k⋅ΞN,kT⋅(ψN,k(s))T]=ψN,k(t)⋅(ψN,k(s))T,
which needs to be equal to the covariance of δN, namely,
(2.35)ψN,k(t)⋅(ψN,k(s))T=μN,k(t)⋅𝚺N,k⋅(μN,k(s))T.
Therefore, since μN,k(mN,k)=Id, we have
(2.36)ψN,k(mN,k)⋅(ψN,k(mN,k))T=𝚺N,k.
We can hence now define ψN,k(mN,k) as a square root σN,k of the symmetric positive matrix ΣN,k, by fixing s=mN,k in (2.35)
(2.37)ψN,k(t)⋅σN,kT=μ(t)⋅σN,k⋅σN,kT.
Eventually, since by assumption we have that ΣN,k is invertible, so is σN,k, and the functions ψN,k can be written as
(2.38)ψN,k(t)=μN,k(t)⋅σN,k
with σN,k being a square root of ΣN,k. Square roots of positive symmetric matrices are uniquely defined up to an orthogonal transformation. Therefore, all square roots of ΣN,k are related by orthogonal transformations σN,k′=σN,k·ON,k, where ON,k·ON,kT=Id. This property immediately extends to the functions ψN,k we are studying: two different functions ψN,k and ψN,k′ satisfying the theorem differ from an orthogonal transformation ON,k. We proved that, for ψN,k(t)·ΞN,k to have the same law as δN(t) in the interval SN,k, the functions ψN,k with support in SN,k are necessarily of the form μN,k(t)·σN,k. It is straightforward to show the sufficient condition that provided such a set of functions, the processes δtN and dtN are equal in law, which ends the proof of the theorem.Using the expressions obtained in Proposition2.1, we can make completely explicit the form of the basis in terms of the functions f,g, and h:(2.39)ψn,k(t)={g(t)g-1(mn,k)hmn,k(ln,k,t)(hmn,k(ln,k,mn,k))-1σn,k,forln,k≤t≤mn,k,g(t)g-1(mn,k)hmn,k(t,rn,k)(hmn,k(mn,k,rn,k))-1σn,k,formn,k≤t≤rn,k,
and σn,k satisfies(2.40)σn,k⋅σn,kT=hmn,k(ln,k,mn,k)(hmn,k(ln,k,rn,k))-1hmn,k(mn,k,rn,k).
Note that σn,k can be defined uniquely as the symmetric positive square root, or as the lower triangular matrix resulting from the Cholesky decomposition of Σn,k.Let us now define the functionψ0,0:[0,1]↦ℝd×d such that the process ψ0,0(t)·Ξ0,0 has the same covariance as Zt0, which is computed using exactly the same technique as that developed in the proof of Theorem 2.4 and that has the expression(2.41)ψ0,0(t)=g(t)h0(l0,0,t)(h0(l0,0,r0,0))-1g-1(r0,0)σ0,0,
for σ0,0, a square root of Cr0,0 the covariance matrix of Xr0,0 which from (2.5) reads(2.42)F(0,1)h0(1,1)F(0,1)T=g(1)h0(1,1)(g(1))T.We are now in position to show the following corollary of Theorem2.4.Corollary 2.5.
The Gauss-Markov processZtN is equal in law to the process
(2.43)XtN=∑n=0N-1∑k∈Inψn,k(t)⋅Ξn,k,
where Ξn,k are independent standard normal random variables 𝒩(0,Id).Proof.
We have(2.44)ZtN=(ZtN-ZtN-1)+(ZtN-1-ZtN-2)+⋯+(Zt2-Zt1)+Zt1=∑n=1N-1δtn+Zt1=∑n=1N-1∑k∈Inψn,k(t)⋅Ξn,k+ψ0,0(t)⋅Ξ0,0=∑n=0N-1∑k∈Inψn,k(t)⋅Ξn,k.We therefore identified a collection of functions{ψn,k}(n,k)∈ℐ that allows a simple construction of the Gauss-Markov process iteratively conditioned on increasingly finer partitions of the interval [0,1]. We will show that this sequence ZtN converges almost surely towards the Gauss-Markov process Xt used to construct the basis, proving that these finite-dimensional continuous functions ZtN form an asymptotically accurate description of the initial process. Beforehand, we rigorously study the Hilbertian properties of the collection of functions we just defined.


## 2.1. Multidimensional Gauss-Markov Processes

After recalling the definition of the multidimensional Gauss-Markov processes in terms of stochastic integral, we use the well-known conditioning formula for the Gaussian vectors to characterize the law of the Gauss-Markov bridge processes.


### 2.1.1. Notations and Definitions

Let(Wt,ℱt,t∈[0,1]) be an m-dimensional Wiener process, consider the continuous functions α:[0,1]→ℝd×dΓ:[0,1]→ℝd×m, and define the positive bounded continuous function Γ=Γ·ΓT:[0,1]→ℝd×d. The d-dimensional Ornstein-Uhlenbeck process associated with these parameters is solution of the equation(2.1)dXt=α(t)⋅Xtdt+Γ(t)⋅dWt,
and with initial condition Xt0 in t0, it reads(2.2)Xt=F(t0,t)⋅Xt0+F(t0,t)⋅∫t0tF(s,t0)⋅Γ(s)⋅dWs,
where F(t0,t) is the flow of the equation, namely, the solution in ℝd×d of the linear equation:(2.3)∂F(t0,t)∂t=α(t)F(t),F(t0,t0)=Id.
Note that the flow F(t0,t) enjoys the chain rule property:(2.4)F(t0,t)=F(t1,t)⋅F(t0,t1).
For all t,s such that t0<s,t, the vectors Xt and Xs admit the covariance(2.5)Ct0(s,t)=F(t0,t)(∫t0t∧sF(w,t0)Γ(w)F(w,t0)Tdw)F(t0,s)T=F(t0,t)ht0(s,t)F(t0,s)T,
where we further defined hu(s,t) the function(2.6)hu(s,t)=∫stF(w,u)⋅Γ(w)⋅F(w,u)Tdw,
which will be of particular interest in the sequel. Note that because of the chain rule property of the flow, we have (2.7)hv(s,t)=F(v,u)hu(s,t)F(v,u)T.
We suppose that the process X is never degenerated, that is, for all t0<u<v, all the components of the vector Xv taking into account Xu are nondeterministic random variables, which is equivalent to saying that the covariance matrix of Xv taking into account Xu, denoted by Cu(v,v) is symmetric positive definite for any u≠v. Therefore, assuming the initial condition X0=0, the multidimensional centered process X has a representation (similar to Doob’s representation for one-dimensional processes, see [35]) of form(2.8)Xt=g(t)∫0tf(s)⋅dWs,
with g(t)=F(0,t) and f(t)=F(t,0)·Γ(t).Note that the processes considered in this paper are defined on the time interval[0,1]. However, because of the time-rescaling property of these processes, considering the processes on this time interval is equivalent to considering the process on any other bounded interval without loss of generality.


### 2.1.2. Conditional Law and Gauss-Markov Bridges

As stated in the introduction, we aim at defining a multiresolution description of Gauss-Markov processes. Such a description can be seen as a multiresolution interpolation of the process that is getting increasingly finer. This principle, in addition to the Markov property, prescribes to characterize the law of the corresponding Gauss-Markov bridge, that is, the Gauss-Markov process under consideration, conditioned on its initial and final values. The bridge process of the Gauss process is still a Gauss process and, for a Markov process, its law can be computed as follows.Proposition 2.1.
Lettx≤tz two times in the interval [0,1]. For any t∈[tx,tz], the random variable Xt conditioned on Xtx=x and Xtz=z is a Gaussian variable with covariance matrix Σ(t) and mean vector μ(t) given by
(2.9)𝚺(t;tx,tz)=ht(tx,t)(ht(tx,tz))-1ht(t,tz),μ(t)=μl(t;tx,tz)⋅x+μr(t;tx,tz)⋅z,
where the continuous matrix functions μl(·;tx,tz) and μr(·;tx,tz) of ℝd×d are given by
(2.10)μl(t;tx,tz)=F(tx,t)htx(t,tz)(htx(tx,tz))-1,μr(t;tx,tz)=F(tz,t)htz(tx,t)(htz(tx,tz))-1.Note that the functionsμl and μr have the property that μl(tx;tx,tz)=μr(tz;tx,tz)=Id and μl(tz;tx,tz)=μr(tx;tx,tz)=0 ensuring that the process is indeed equal to x at time tx and z at time tz.Proof.
Lettx,tz be two times of the interval [0,1] such that tx<tz, and let t∈[tx,tz]. We consider the Gaussian random variable ξ=(Xt,Xtz) conditioned on the fact that Xtx=x. Its mean can be easily computed from   expression (2.2) and reads
(2.11)(mt,mtz)=(F(tx,t)x,F(tx,tz)x)=(g(t)g-1(tx)x,g(tz)g-1(tx)x),
and its covariance matrix, from (2.5), reads
(2.12)[Ct,tCt,tzCtz,tCtz,tz]=[F(tx,t)htx(tx,t)F(tx,t)TF(tx,t)htx(tx,t)F(tx,tz)TF(tx,tz)htx(tx,t)F(tx,t)TF(tx,tz)htx(tx,tz)F(tx,tz)T]=[ht(tx,t)ht(tx,t)F(t,tz)TF(t,tz)ht(tx,t)F(t,tz)ht(tx,tz)F(t,tz)T].
From there, we apply the conditioning formula for the Gaussian vectors (see, e.g., [36]) to infer the law of Xt conditioned on Xtx=x and Xtz=z, that is the law 𝒩(μ(t),Σ(t;tx,tz)) of Bt where B denotes the bridge process obtained by pinning X in tx and tz. The covariance matrix is given by
(2.13)𝚺(t;tx,tz)=Cy,y-Cy,zCz,z-1Cz,y=ht(tx,t)-ht(tx,t)(ht(tx,tz))-1ht(tx,t)=ht(tx,t)(ht(tx,tz))-1ht(t,tz),
and the mean reads
(2.14)μ(t)=my+Cy,zCz,z-1(z-mz)=F(tx,t)(Id-htx(tx,t)(htx(tx,tz))-1)x+F(tz,t)htz(tx,t)(htz(tx,tz))-1z,=F(tx,t)htx(t,tz)(htx(tx,tz))-1︸μl(t;tx,tz)⋅x+F(tz,t)htz(tx,t)(htz(tx,tz))-1︸μr(t;tx,tz)⋅z,
where we have used the fact that htx(tx,tz)=htz(tx,t)+htx(t,tz). The regularity of the thus-defined functions μx and μz directly stems from the regularity of the flow operator F. Moreover, since for any 0≤t,u≤1, we observe that F(t,t)=Id and hu(t,t)=0; we clearly have μx(tx)=μy(t)=Id and μx(t)=μy(tx)=0.Remark 2.2.
Note that these laws can also be computed using the expression of the density of the processes  but involve  more intricate calculations. An alternative approach also provides a representation of Gauss-Markov bridges with the use of integral and anticipative representation [37]. These approaches allow to compute the probability distribution of the Gauss-Markov bridge as a process (i.e., allows to compute the covariances), but since this will be of no use in the sequel, we do not provide the expressions.


## 2.1.1. Notations and Definitions

Let(Wt,ℱt,t∈[0,1]) be an m-dimensional Wiener process, consider the continuous functions α:[0,1]→ℝd×dΓ:[0,1]→ℝd×m, and define the positive bounded continuous function Γ=Γ·ΓT:[0,1]→ℝd×d. The d-dimensional Ornstein-Uhlenbeck process associated with these parameters is solution of the equation(2.1)dXt=α(t)⋅Xtdt+Γ(t)⋅dWt,
and with initial condition Xt0 in t0, it reads(2.2)Xt=F(t0,t)⋅Xt0+F(t0,t)⋅∫t0tF(s,t0)⋅Γ(s)⋅dWs,
where F(t0,t) is the flow of the equation, namely, the solution in ℝd×d of the linear equation:(2.3)∂F(t0,t)∂t=α(t)F(t),F(t0,t0)=Id.
Note that the flow F(t0,t) enjoys the chain rule property:(2.4)F(t0,t)=F(t1,t)⋅F(t0,t1).
For all t,s such that t0<s,t, the vectors Xt and Xs admit the covariance(2.5)Ct0(s,t)=F(t0,t)(∫t0t∧sF(w,t0)Γ(w)F(w,t0)Tdw)F(t0,s)T=F(t0,t)ht0(s,t)F(t0,s)T,
where we further defined hu(s,t) the function(2.6)hu(s,t)=∫stF(w,u)⋅Γ(w)⋅F(w,u)Tdw,
which will be of particular interest in the sequel. Note that because of the chain rule property of the flow, we have (2.7)hv(s,t)=F(v,u)hu(s,t)F(v,u)T.
We suppose that the process X is never degenerated, that is, for all t0<u<v, all the components of the vector Xv taking into account Xu are nondeterministic random variables, which is equivalent to saying that the covariance matrix of Xv taking into account Xu, denoted by Cu(v,v) is symmetric positive definite for any u≠v. Therefore, assuming the initial condition X0=0, the multidimensional centered process X has a representation (similar to Doob’s representation for one-dimensional processes, see [35]) of form(2.8)Xt=g(t)∫0tf(s)⋅dWs,
with g(t)=F(0,t) and f(t)=F(t,0)·Γ(t).Note that the processes considered in this paper are defined on the time interval[0,1]. However, because of the time-rescaling property of these processes, considering the processes on this time interval is equivalent to considering the process on any other bounded interval without loss of generality.


## 2.1.2. Conditional Law and Gauss-Markov Bridges

As stated in the introduction, we aim at defining a multiresolution description of Gauss-Markov processes. Such a description can be seen as a multiresolution interpolation of the process that is getting increasingly finer. This principle, in addition to the Markov property, prescribes to characterize the law of the corresponding Gauss-Markov bridge, that is, the Gauss-Markov process under consideration, conditioned on its initial and final values. The bridge process of the Gauss process is still a Gauss process and, for a Markov process, its law can be computed as follows.Proposition 2.1.
Lettx≤tz two times in the interval [0,1]. For any t∈[tx,tz], the random variable Xt conditioned on Xtx=x and Xtz=z is a Gaussian variable with covariance matrix Σ(t) and mean vector μ(t) given by
(2.9)𝚺(t;tx,tz)=ht(tx,t)(ht(tx,tz))-1ht(t,tz),μ(t)=μl(t;tx,tz)⋅x+μr(t;tx,tz)⋅z,
where the continuous matrix functions μl(·;tx,tz) and μr(·;tx,tz) of ℝd×d are given by
(2.10)μl(t;tx,tz)=F(tx,t)htx(t,tz)(htx(tx,tz))-1,μr(t;tx,tz)=F(tz,t)htz(tx,t)(htz(tx,tz))-1.Note that the functionsμl and μr have the property that μl(tx;tx,tz)=μr(tz;tx,tz)=Id and μl(tz;tx,tz)=μr(tx;tx,tz)=0 ensuring that the process is indeed equal to x at time tx and z at time tz.Proof.
Lettx,tz be two times of the interval [0,1] such that tx<tz, and let t∈[tx,tz]. We consider the Gaussian random variable ξ=(Xt,Xtz) conditioned on the fact that Xtx=x. Its mean can be easily computed from   expression (2.2) and reads
(2.11)(mt,mtz)=(F(tx,t)x,F(tx,tz)x)=(g(t)g-1(tx)x,g(tz)g-1(tx)x),
and its covariance matrix, from (2.5), reads
(2.12)[Ct,tCt,tzCtz,tCtz,tz]=[F(tx,t)htx(tx,t)F(tx,t)TF(tx,t)htx(tx,t)F(tx,tz)TF(tx,tz)htx(tx,t)F(tx,t)TF(tx,tz)htx(tx,tz)F(tx,tz)T]=[ht(tx,t)ht(tx,t)F(t,tz)TF(t,tz)ht(tx,t)F(t,tz)ht(tx,tz)F(t,tz)T].
From there, we apply the conditioning formula for the Gaussian vectors (see, e.g., [36]) to infer the law of Xt conditioned on Xtx=x and Xtz=z, that is the law 𝒩(μ(t),Σ(t;tx,tz)) of Bt where B denotes the bridge process obtained by pinning X in tx and tz. The covariance matrix is given by
(2.13)𝚺(t;tx,tz)=Cy,y-Cy,zCz,z-1Cz,y=ht(tx,t)-ht(tx,t)(ht(tx,tz))-1ht(tx,t)=ht(tx,t)(ht(tx,tz))-1ht(t,tz),
and the mean reads
(2.14)μ(t)=my+Cy,zCz,z-1(z-mz)=F(tx,t)(Id-htx(tx,t)(htx(tx,tz))-1)x+F(tz,t)htz(tx,t)(htz(tx,tz))-1z,=F(tx,t)htx(t,tz)(htx(tx,tz))-1︸μl(t;tx,tz)⋅x+F(tz,t)htz(tx,t)(htz(tx,tz))-1︸μr(t;tx,tz)⋅z,
where we have used the fact that htx(tx,tz)=htz(tx,t)+htx(t,tz). The regularity of the thus-defined functions μx and μz directly stems from the regularity of the flow operator F. Moreover, since for any 0≤t,u≤1, we observe that F(t,t)=Id and hu(t,t)=0; we clearly have μx(tx)=μy(t)=Id and μx(t)=μy(tx)=0.Remark 2.2.
Note that these laws can also be computed using the expression of the density of the processes  but involve  more intricate calculations. An alternative approach also provides a representation of Gauss-Markov bridges with the use of integral and anticipative representation [37]. These approaches allow to compute the probability distribution of the Gauss-Markov bridge as a process (i.e., allows to compute the covariances), but since this will be of no use in the sequel, we do not provide the expressions.


## 2.2. The Multiresolution Description of Gauss-Markov Processes

Recognizing the Gauss property and the Markov property as the two crucial elements for a stochastic process to be expanded to Lévy-Cesielski, our approach first proposes to exhibit bases of deterministic functions that would play the role of the Schauder bases for the Wiener process. In this regard, we first expect such functions to be continuous and compactly supported on increasingly finer supports (i.e., subintervals of the definition interval[0,1]) in a similar nested binary tree structure. Then, as in the Lévy-Ciesielski construction, we envision that, at each resolution (i.e., on each support), the partially constructed process (up to the resolution of the support) has the same conditional expectation as the Gauss-Markov process when conditioned on the endpoints of the supports. The partial sums obtained with independent Gaussian coefficients of law 𝒩(0,1) will thus approximate the targeted Gauss-Markov process in a multiresolution fashion, in the sense that, at every resolution, considering these two processes on the interval  endpoints yields finite-dimensional Gaussian vectors of the same law.


### 2.2.1. Nested Structure of the Sequence of Supports

Here, we define the nested sequence of segments that constitute the supports of the multiresolution basis. We construct such a sequence by recursively partitioning the interval[0,1].More precisely, starting fromS1,0=[l1,0,r1,0] with l1,0=0 and r1,0=1, we iteratively apply the following operation. Suppose that, at the nth step, the interval [0,1] is decomposed into 2n-1 intervals Sn,k=[ln,k,rn,k], called supports, such that ln,k+1=rn,k for 0≤k<2n-1. Each of these intervals is then subdivided into two child intervals, a left-child Sn+1,2k and a right-child Sn+1,2k+1, and the subdivision point rn+1,2k=ln+1,2k+1 is denoted by mn,k. Therefore, we have defined three sequences of real ln,k, mn,k, and rn,k for n>0 and 0≤k<2n-1 satisfying l0,0=0≤ln,k<mn,k<rn,k≤r0,0=1 and(2.15)ln+1,2k=ln,k,mn,k=rn+1,2k=ln+1,2k+1,rn+1,2k+1=rn,k
with the convention   l0,0=0 and r0,0=1 and S0,0=[0,1]. The resulting sequence of supports {Sn,k;n≤0,0≤k<2n-1} clearly has a binary tree structure.For the sake of compactness of notations, we defineℐ the set of indices(2.16)I=⋃n<NInwithIN={(n,k)∈N2∣0<n≤N,0≤k<2n-1},
and for N>0, we define DN={mn,k,(n,k)∈ℐN-1}∪{0,1}, the set of endpoints of the intervals SN,k. We additionally require that there exists ρ∈(0,1) such that for all (n,k)∈ℐmax(rn,k-mn,k,mn,k-ln,k)<ρ(rn,k-ln,k) which in particular implies that(2.17)lim⁡n→∞sup⁡krn,k-ln,k=0
and ensures that the set of endpoints ∪N∈NDN is everywhere dense in [0,1]. The simplest case of such partitions is the dyadic partition of [0,1], where the endpoints for (n,k)∈ℐ read(2.18)ln,k=k2-n+1,mn,k=(2k+1)2-n,rn,k=(k+1)2-n+1,
in which case the endpoints are simply the dyadic points ∪NDN={k2-N|0≤k≤2N}.  Figure 1 represents the global architecture of the nested sequence of intervals.Figure 1
A sequence of nested intervals.The nested structure of the supports, together with the constraint of continuity of the bases elements, implies that only a finite number of coefficients are needed to construct the exact value of the process at a given endpoint, thus providing us with an exact schema to simulate the sample values of the process on the endpoint up to an arbitrary resolution, as we will further explore.


### 2.2.2. Innovation Processes for Gauss-Markov Processes

ForXt, a multidimensional Gauss-Markov process, we call the multiresolution description of a process the sequence of conditional expectations on the nested sets of endpoints Dn. In detail, if we denote by ℱN the filtration generated by {Xt;t∈DN} given the values of the process at the endpoints DN of the partition, we introduce the sequence of the Gaussian processes (ZtN)N≥1 defined by:(2.19)ZtN=E[Xt∣FN]=EN[Xt].
These processes ZN can be naturally viewed as an interpolation of the process X sampled at the increasingly finer times DN  since for all t∈DN we have ZtN=XtN. The innovation process (δtN,ℱt,t∈[0,1]) is defined as the update transforming the process ZtN into ZtN+1, that is,
(2.20)δtN=ZtN+1-ZtN.
It corresponds to the difference the additional knowledge of the process at the points mN,k make on the conditional expectation of the process. This process satisfies the following important properties that found our multiresolution construction.Proposition 2.3.
The innovation processδtN is a centered Gaussian process independent of the processes Ztn for any n≤N. For s∈SN,k and t∈SN,p with k,p∈ℐN, the covariance of the innovation process reads
(2.21)EN[δtN⋅(δsN)T]={μN,k(t)⋅𝚺N,k⋅μN,k(t)Tifk=p,0ifk≠p,
where
(2.22)μN,k(t)={μr(t;lN,k,mN,k),t∈[lN,k,mN,k],μl(t;mN,k,rN,k),t∈[mN,k,rN,k].
with μl, μr and ΣN,k=Σ(mN,k;lN,k,rN,k) as defined in Proposition 2.1.Proof.
Because of the Markovian property of the processX, the law of the process ZN can be computed from the bridge formula derived in Proposition 2.1 and we have
(2.23)ZtN=μl(t;lN,k,rN,k)⋅XlN,k+μr(t;lN,k,rN,k)⋅XrN,k,ZtN+1={μl(t;lN,k,mN,k)⋅XlN,k+μr(t;lN,k,mN,k)⋅XmN,k,fort∈[lN,k,mN,k],μl(t;mN,k,rN,k)⋅XmN,k+μr(t;mN,k,rN,k)⋅XrN,k,fort∈[lN,k,mN,k].
Therefore, the innovation process can be written for t∈SN,k as
(2.24)δtN=μN,kN(t)⋅XmN,k+νN(t)⋅QtN,
where QtN is a ℱN measurable process  νN(t) a deterministic matrix function and
(2.25)μN,k(t)={μr(t;lN,k,mN,k),t∈[lN,k,mN,k],μl(t;mN,k,rN,k),t∈[mN,k,rN,k].
The expressions of ν and Q are quite complex  but are highly simplified when noting that
(2.26)E[δtN∣FN]=E[ZtN+1∣FN]-ZtN=E[E[Zt∣FN+1]∣FN]-ZtN=0
directly implying that ν(t)·QtN=μN(t)·ZmN,kN and yielding the remarkably compact expression
(2.27)δtN=μN,k(t)⋅(XmN,k-ZmN,kN).
This process is a centered Gaussian process. Moreover, observing that it is ℱN-measurable, it can be written as
(2.28)δtN=μN,k(t)⋅({XmN,k∣FN}-ZmN,kN),
and the process {XmN,k∣ℱN} appears as the Gauss-Markov bridge conditioned at times lN,k and rN,k, and whose covariance is given by Proposition 2.1 and that has the expression
(2.29)𝚺N,k=𝚺(mN,k;lN,k,rN,k)=hmn,k(ln,k,mn,k)(hmn,k(ln,k,rn,k))-1hmn,k(mn,k,rn,k).
Let (s,t)∈[0,1]2, and assume that s∈SN,k and t∈SN,p. If k≠p, then, because of the Markov property of the process X, the two bridges are independent and therefore the covariance 𝔼N[δtN·(δsN)T] is zero. If k=p, we have
(2.30)EN[δtN⋅(δsN)T]=μN,k(t)⋅𝚺N,k⋅μN,k(s)T.
Eventually, the independence property stems from the simple properties of the conditional expectation. Indeed, let n≤N. We have
(2.31)E[Ztn⋅(δsN)T]=E[Ztn⋅(ZsN+1-ZsN)T]=E[E[Xt∣Fn]⋅(E[XsT∣FN+1]-E[XsT∣FN])]=E[E[Xt∣Fn]⋅E[XsT∣FN+1]]-E[E[Xt∣Fn]⋅E[XsT∣FN]]=E[Ztn(Zsn)T]-E[Ztn(Zsn)T]=0
and the fact that a zero covariance between two Gaussian processes implies the independence of these processes concludes the proof.


### 2.2.3. Derivation of the Candidate Multiresolution Bases of Functions

We deduce from the previous proposition the following fundamental theorem of this paper.Theorem 2.4.
For allN∈ℕ, there exists a collection of   ψN,k:[0,1]↦ℝd×d that are zero outside the subinterval SN,k   such that in distribution one has:
(2.32)δtN=∑k∈INψN,k(t)⋅ΞN,k,
where ΞN,k are independent d-dimensional standard normal random variables (i.e., of law 𝒩(0,Id)). This basis of functions is unique up to an orthogonal transformation.Proof.
The two processesδtN and dtN=def∑k∈ℐNψN,k(t)·ΞN,k are two Gaussian processes of mean zero. Therefore, we are searching for functions ψN,k vanishing outside SN,k and ensuring that the two processes have the same probability distribution. A necessary and sufficient condition for the two processes to have the same probability distribution is to have the same covariance function (see, e.g., [36]). We therefore need to show the existence of a collection of functions ψN,k(t) functions that vanish outside the subinterval SN,k and that ensure that the covariance of the process dN is equal to the covariance of δN. Let (s,t)∈[0,1] such that s∈SN,k and t∈SN,p. If k≠p, the assumption fact that the functions ψN,k vanish outside SN,k implies that
(2.33)E[dtN⋅(dsN)T]=0.
If k=p, the covariance reads
(2.34)E[dtN⋅(dsN)T]=E[ψN,k(t)⋅ΞN,k⋅ΞN,kT⋅(ψN,k(s))T]=ψN,k(t)⋅(ψN,k(s))T,
which needs to be equal to the covariance of δN, namely,
(2.35)ψN,k(t)⋅(ψN,k(s))T=μN,k(t)⋅𝚺N,k⋅(μN,k(s))T.
Therefore, since μN,k(mN,k)=Id, we have
(2.36)ψN,k(mN,k)⋅(ψN,k(mN,k))T=𝚺N,k.
We can hence now define ψN,k(mN,k) as a square root σN,k of the symmetric positive matrix ΣN,k, by fixing s=mN,k in (2.35)
(2.37)ψN,k(t)⋅σN,kT=μ(t)⋅σN,k⋅σN,kT.
Eventually, since by assumption we have that ΣN,k is invertible, so is σN,k, and the functions ψN,k can be written as
(2.38)ψN,k(t)=μN,k(t)⋅σN,k
with σN,k being a square root of ΣN,k. Square roots of positive symmetric matrices are uniquely defined up to an orthogonal transformation. Therefore, all square roots of ΣN,k are related by orthogonal transformations σN,k′=σN,k·ON,k, where ON,k·ON,kT=Id. This property immediately extends to the functions ψN,k we are studying: two different functions ψN,k and ψN,k′ satisfying the theorem differ from an orthogonal transformation ON,k. We proved that, for ψN,k(t)·ΞN,k to have the same law as δN(t) in the interval SN,k, the functions ψN,k with support in SN,k are necessarily of the form μN,k(t)·σN,k. It is straightforward to show the sufficient condition that provided such a set of functions, the processes δtN and dtN are equal in law, which ends the proof of the theorem.Using the expressions obtained in Proposition2.1, we can make completely explicit the form of the basis in terms of the functions f,g, and h:(2.39)ψn,k(t)={g(t)g-1(mn,k)hmn,k(ln,k,t)(hmn,k(ln,k,mn,k))-1σn,k,forln,k≤t≤mn,k,g(t)g-1(mn,k)hmn,k(t,rn,k)(hmn,k(mn,k,rn,k))-1σn,k,formn,k≤t≤rn,k,
and σn,k satisfies(2.40)σn,k⋅σn,kT=hmn,k(ln,k,mn,k)(hmn,k(ln,k,rn,k))-1hmn,k(mn,k,rn,k).
Note that σn,k can be defined uniquely as the symmetric positive square root, or as the lower triangular matrix resulting from the Cholesky decomposition of Σn,k.Let us now define the functionψ0,0:[0,1]↦ℝd×d such that the process ψ0,0(t)·Ξ0,0 has the same covariance as Zt0, which is computed using exactly the same technique as that developed in the proof of Theorem 2.4 and that has the expression(2.41)ψ0,0(t)=g(t)h0(l0,0,t)(h0(l0,0,r0,0))-1g-1(r0,0)σ0,0,
for σ0,0, a square root of Cr0,0 the covariance matrix of Xr0,0 which from (2.5) reads(2.42)F(0,1)h0(1,1)F(0,1)T=g(1)h0(1,1)(g(1))T.We are now in position to show the following corollary of Theorem2.4.Corollary 2.5.
The Gauss-Markov processZtN is equal in law to the process
(2.43)XtN=∑n=0N-1∑k∈Inψn,k(t)⋅Ξn,k,
where Ξn,k are independent standard normal random variables 𝒩(0,Id).Proof.
We have(2.44)ZtN=(ZtN-ZtN-1)+(ZtN-1-ZtN-2)+⋯+(Zt2-Zt1)+Zt1=∑n=1N-1δtn+Zt1=∑n=1N-1∑k∈Inψn,k(t)⋅Ξn,k+ψ0,0(t)⋅Ξ0,0=∑n=0N-1∑k∈Inψn,k(t)⋅Ξn,k.We therefore identified a collection of functions{ψn,k}(n,k)∈ℐ that allows a simple construction of the Gauss-Markov process iteratively conditioned on increasingly finer partitions of the interval [0,1]. We will show that this sequence ZtN converges almost surely towards the Gauss-Markov process Xt used to construct the basis, proving that these finite-dimensional continuous functions ZtN form an asymptotically accurate description of the initial process. Beforehand, we rigorously study the Hilbertian properties of the collection of functions we just defined.


## 2.2.1. Nested Structure of the Sequence of Supports

Here, we define the nested sequence of segments that constitute the supports of the multiresolution basis. We construct such a sequence by recursively partitioning the interval[0,1].More precisely, starting fromS1,0=[l1,0,r1,0] with l1,0=0 and r1,0=1, we iteratively apply the following operation. Suppose that, at the nth step, the interval [0,1] is decomposed into 2n-1 intervals Sn,k=[ln,k,rn,k], called supports, such that ln,k+1=rn,k for 0≤k<2n-1. Each of these intervals is then subdivided into two child intervals, a left-child Sn+1,2k and a right-child Sn+1,2k+1, and the subdivision point rn+1,2k=ln+1,2k+1 is denoted by mn,k. Therefore, we have defined three sequences of real ln,k, mn,k, and rn,k for n>0 and 0≤k<2n-1 satisfying l0,0=0≤ln,k<mn,k<rn,k≤r0,0=1 and(2.15)ln+1,2k=ln,k,mn,k=rn+1,2k=ln+1,2k+1,rn+1,2k+1=rn,k
with the convention   l0,0=0 and r0,0=1 and S0,0=[0,1]. The resulting sequence of supports {Sn,k;n≤0,0≤k<2n-1} clearly has a binary tree structure.For the sake of compactness of notations, we defineℐ the set of indices(2.16)I=⋃n<NInwithIN={(n,k)∈N2∣0<n≤N,0≤k<2n-1},
and for N>0, we define DN={mn,k,(n,k)∈ℐN-1}∪{0,1}, the set of endpoints of the intervals SN,k. We additionally require that there exists ρ∈(0,1) such that for all (n,k)∈ℐmax(rn,k-mn,k,mn,k-ln,k)<ρ(rn,k-ln,k) which in particular implies that(2.17)lim⁡n→∞sup⁡krn,k-ln,k=0
and ensures that the set of endpoints ∪N∈NDN is everywhere dense in [0,1]. The simplest case of such partitions is the dyadic partition of [0,1], where the endpoints for (n,k)∈ℐ read(2.18)ln,k=k2-n+1,mn,k=(2k+1)2-n,rn,k=(k+1)2-n+1,
in which case the endpoints are simply the dyadic points ∪NDN={k2-N|0≤k≤2N}.  Figure 1 represents the global architecture of the nested sequence of intervals.Figure 1
A sequence of nested intervals.The nested structure of the supports, together with the constraint of continuity of the bases elements, implies that only a finite number of coefficients are needed to construct the exact value of the process at a given endpoint, thus providing us with an exact schema to simulate the sample values of the process on the endpoint up to an arbitrary resolution, as we will further explore.


## 2.2.2. Innovation Processes for Gauss-Markov Processes

ForXt, a multidimensional Gauss-Markov process, we call the multiresolution description of a process the sequence of conditional expectations on the nested sets of endpoints Dn. In detail, if we denote by ℱN the filtration generated by {Xt;t∈DN} given the values of the process at the endpoints DN of the partition, we introduce the sequence of the Gaussian processes (ZtN)N≥1 defined by:(2.19)ZtN=E[Xt∣FN]=EN[Xt].
These processes ZN can be naturally viewed as an interpolation of the process X sampled at the increasingly finer times DN  since for all t∈DN we have ZtN=XtN. The innovation process (δtN,ℱt,t∈[0,1]) is defined as the update transforming the process ZtN into ZtN+1, that is,
(2.20)δtN=ZtN+1-ZtN.
It corresponds to the difference the additional knowledge of the process at the points mN,k make on the conditional expectation of the process. This process satisfies the following important properties that found our multiresolution construction.Proposition 2.3.
The innovation processδtN is a centered Gaussian process independent of the processes Ztn for any n≤N. For s∈SN,k and t∈SN,p with k,p∈ℐN, the covariance of the innovation process reads
(2.21)EN[δtN⋅(δsN)T]={μN,k(t)⋅𝚺N,k⋅μN,k(t)Tifk=p,0ifk≠p,
where
(2.22)μN,k(t)={μr(t;lN,k,mN,k),t∈[lN,k,mN,k],μl(t;mN,k,rN,k),t∈[mN,k,rN,k].
with μl, μr and ΣN,k=Σ(mN,k;lN,k,rN,k) as defined in Proposition 2.1.Proof.
Because of the Markovian property of the processX, the law of the process ZN can be computed from the bridge formula derived in Proposition 2.1 and we have
(2.23)ZtN=μl(t;lN,k,rN,k)⋅XlN,k+μr(t;lN,k,rN,k)⋅XrN,k,ZtN+1={μl(t;lN,k,mN,k)⋅XlN,k+μr(t;lN,k,mN,k)⋅XmN,k,fort∈[lN,k,mN,k],μl(t;mN,k,rN,k)⋅XmN,k+μr(t;mN,k,rN,k)⋅XrN,k,fort∈[lN,k,mN,k].
Therefore, the innovation process can be written for t∈SN,k as
(2.24)δtN=μN,kN(t)⋅XmN,k+νN(t)⋅QtN,
where QtN is a ℱN measurable process  νN(t) a deterministic matrix function and
(2.25)μN,k(t)={μr(t;lN,k,mN,k),t∈[lN,k,mN,k],μl(t;mN,k,rN,k),t∈[mN,k,rN,k].
The expressions of ν and Q are quite complex  but are highly simplified when noting that
(2.26)E[δtN∣FN]=E[ZtN+1∣FN]-ZtN=E[E[Zt∣FN+1]∣FN]-ZtN=0
directly implying that ν(t)·QtN=μN(t)·ZmN,kN and yielding the remarkably compact expression
(2.27)δtN=μN,k(t)⋅(XmN,k-ZmN,kN).
This process is a centered Gaussian process. Moreover, observing that it is ℱN-measurable, it can be written as
(2.28)δtN=μN,k(t)⋅({XmN,k∣FN}-ZmN,kN),
and the process {XmN,k∣ℱN} appears as the Gauss-Markov bridge conditioned at times lN,k and rN,k, and whose covariance is given by Proposition 2.1 and that has the expression
(2.29)𝚺N,k=𝚺(mN,k;lN,k,rN,k)=hmn,k(ln,k,mn,k)(hmn,k(ln,k,rn,k))-1hmn,k(mn,k,rn,k).
Let (s,t)∈[0,1]2, and assume that s∈SN,k and t∈SN,p. If k≠p, then, because of the Markov property of the process X, the two bridges are independent and therefore the covariance 𝔼N[δtN·(δsN)T] is zero. If k=p, we have
(2.30)EN[δtN⋅(δsN)T]=μN,k(t)⋅𝚺N,k⋅μN,k(s)T.
Eventually, the independence property stems from the simple properties of the conditional expectation. Indeed, let n≤N. We have
(2.31)E[Ztn⋅(δsN)T]=E[Ztn⋅(ZsN+1-ZsN)T]=E[E[Xt∣Fn]⋅(E[XsT∣FN+1]-E[XsT∣FN])]=E[E[Xt∣Fn]⋅E[XsT∣FN+1]]-E[E[Xt∣Fn]⋅E[XsT∣FN]]=E[Ztn(Zsn)T]-E[Ztn(Zsn)T]=0
and the fact that a zero covariance between two Gaussian processes implies the independence of these processes concludes the proof.


## 2.2.3. Derivation of the Candidate Multiresolution Bases of Functions

We deduce from the previous proposition the following fundamental theorem of this paper.Theorem 2.4.
For allN∈ℕ, there exists a collection of   ψN,k:[0,1]↦ℝd×d that are zero outside the subinterval SN,k   such that in distribution one has:
(2.32)δtN=∑k∈INψN,k(t)⋅ΞN,k,
where ΞN,k are independent d-dimensional standard normal random variables (i.e., of law 𝒩(0,Id)). This basis of functions is unique up to an orthogonal transformation.Proof.
The two processesδtN and dtN=def∑k∈ℐNψN,k(t)·ΞN,k are two Gaussian processes of mean zero. Therefore, we are searching for functions ψN,k vanishing outside SN,k and ensuring that the two processes have the same probability distribution. A necessary and sufficient condition for the two processes to have the same probability distribution is to have the same covariance function (see, e.g., [36]). We therefore need to show the existence of a collection of functions ψN,k(t) functions that vanish outside the subinterval SN,k and that ensure that the covariance of the process dN is equal to the covariance of δN. Let (s,t)∈[0,1] such that s∈SN,k and t∈SN,p. If k≠p, the assumption fact that the functions ψN,k vanish outside SN,k implies that
(2.33)E[dtN⋅(dsN)T]=0.
If k=p, the covariance reads
(2.34)E[dtN⋅(dsN)T]=E[ψN,k(t)⋅ΞN,k⋅ΞN,kT⋅(ψN,k(s))T]=ψN,k(t)⋅(ψN,k(s))T,
which needs to be equal to the covariance of δN, namely,
(2.35)ψN,k(t)⋅(ψN,k(s))T=μN,k(t)⋅𝚺N,k⋅(μN,k(s))T.
Therefore, since μN,k(mN,k)=Id, we have
(2.36)ψN,k(mN,k)⋅(ψN,k(mN,k))T=𝚺N,k.
We can hence now define ψN,k(mN,k) as a square root σN,k of the symmetric positive matrix ΣN,k, by fixing s=mN,k in (2.35)
(2.37)ψN,k(t)⋅σN,kT=μ(t)⋅σN,k⋅σN,kT.
Eventually, since by assumption we have that ΣN,k is invertible, so is σN,k, and the functions ψN,k can be written as
(2.38)ψN,k(t)=μN,k(t)⋅σN,k
with σN,k being a square root of ΣN,k. Square roots of positive symmetric matrices are uniquely defined up to an orthogonal transformation. Therefore, all square roots of ΣN,k are related by orthogonal transformations σN,k′=σN,k·ON,k, where ON,k·ON,kT=Id. This property immediately extends to the functions ψN,k we are studying: two different functions ψN,k and ψN,k′ satisfying the theorem differ from an orthogonal transformation ON,k. We proved that, for ψN,k(t)·ΞN,k to have the same law as δN(t) in the interval SN,k, the functions ψN,k with support in SN,k are necessarily of the form μN,k(t)·σN,k. It is straightforward to show the sufficient condition that provided such a set of functions, the processes δtN and dtN are equal in law, which ends the proof of the theorem.Using the expressions obtained in Proposition2.1, we can make completely explicit the form of the basis in terms of the functions f,g, and h:(2.39)ψn,k(t)={g(t)g-1(mn,k)hmn,k(ln,k,t)(hmn,k(ln,k,mn,k))-1σn,k,forln,k≤t≤mn,k,g(t)g-1(mn,k)hmn,k(t,rn,k)(hmn,k(mn,k,rn,k))-1σn,k,formn,k≤t≤rn,k,
and σn,k satisfies(2.40)σn,k⋅σn,kT=hmn,k(ln,k,mn,k)(hmn,k(ln,k,rn,k))-1hmn,k(mn,k,rn,k).
Note that σn,k can be defined uniquely as the symmetric positive square root, or as the lower triangular matrix resulting from the Cholesky decomposition of Σn,k.Let us now define the functionψ0,0:[0,1]↦ℝd×d such that the process ψ0,0(t)·Ξ0,0 has the same covariance as Zt0, which is computed using exactly the same technique as that developed in the proof of Theorem 2.4 and that has the expression(2.41)ψ0,0(t)=g(t)h0(l0,0,t)(h0(l0,0,r0,0))-1g-1(r0,0)σ0,0,
for σ0,0, a square root of Cr0,0 the covariance matrix of Xr0,0 which from (2.5) reads(2.42)F(0,1)h0(1,1)F(0,1)T=g(1)h0(1,1)(g(1))T.We are now in position to show the following corollary of Theorem2.4.Corollary 2.5.
The Gauss-Markov processZtN is equal in law to the process
(2.43)XtN=∑n=0N-1∑k∈Inψn,k(t)⋅Ξn,k,
where Ξn,k are independent standard normal random variables 𝒩(0,Id).Proof.
We have(2.44)ZtN=(ZtN-ZtN-1)+(ZtN-1-ZtN-2)+⋯+(Zt2-Zt1)+Zt1=∑n=1N-1δtn+Zt1=∑n=1N-1∑k∈Inψn,k(t)⋅Ξn,k+ψ0,0(t)⋅Ξ0,0=∑n=0N-1∑k∈Inψn,k(t)⋅Ξn,k.We therefore identified a collection of functions{ψn,k}(n,k)∈ℐ that allows a simple construction of the Gauss-Markov process iteratively conditioned on increasingly finer partitions of the interval [0,1]. We will show that this sequence ZtN converges almost surely towards the Gauss-Markov process Xt used to construct the basis, proving that these finite-dimensional continuous functions ZtN form an asymptotically accurate description of the initial process. Beforehand, we rigorously study the Hilbertian properties of the collection of functions we just defined.


## 3. The Multiresolution Schauder Basis Framework

The above analysis motivates the introduction of a set of functions{ψn,k}(n,k)∈ℐ we now study in details. In particular, we enlighten the structure of the collection of functions ψn,k as a Schauder basis in a certain space 𝒳 of continuous functions from [0,1] to ℝd. The Schauder structure was defined in [38, 39], and its essential characterization is the unique decomposition property: namely that every element x in 𝒳 can be written as a well-formed linear combination(3.1)x=∑(n,k)∈Iψn,k⋅ξn,k,
and that the coefficients satisfying the previous relation are unique.


### 3.1. System of Dual Bases

To complete this program, we need to introduce some quantities that will play a crucial role in expressing the familyψn,k as a Schauder basis for some given space. In   (2.39), two constant matrices ℝd×d appear  that will have a particular importance in the sequel for (n,k) in ℐ with n≠0:(3.2)Ln,k=gT(mn,k)(hmn,k(ln,k,mn,k))-1σn,k=(h(ln,k,mn,k))-1g-1(mn,k)σn,k,Rn,k=gT(mn,k)(hmn,k(mn,k,rn,k))-1σn,k=(h(mn,k,rn,k))-1g-1(mn,k)σn,k,
where h stands for h0. We further define the matrix(3.3)Mn,k=gT(mn,k)σn,k-1T
and we recall that σn,k is a square root of Σn,k, the covariance matrix of Xmn,k, conditionally to Xln,k and Xrn,k, given in (2.29). We stress that the matrices Ln,k,Rn,k, Mn,k, and Σn,k are all invertible and satisfy the important following properties.Proposition 3.1.
For all(n,k) in ℐ, n≠0, one has:(i)
Mn,k=Ln,k+Rn,k(ii)
Σn,k-1=(hmn,k(ln,k,mn,k))-1+(hmn,k(mn,k,rn,k))-1.To prove this proposition, we first establish the following simple lemma of linear algebra.Lemma 3.2.
Given two invertible matricesA and B in GLn(ℝ) such that C=A+B is also invertible, if one defines D=AC-1B, one has the following properties:(i)
D=AC-1B=BC-1A(ii)
D-1=A-1+B-1.Proof.
(i)
D=AC-1B=(C-B)C-1B=B-BC-1B=B(I-C-1B)=BC-1(C-B)=BC-1A.(ii)
(A-1+B-1)D=A-1D+B-1D=A-1AC-1B+B-1BC-1A=C-1(B+A)=C-1C=I.Proof of Proposition3.1.
(ii)
Directly stems from Lemma3.2, item (ii) by posing A=hmn,k(ln,k,mn,k), B=hmn,k(mn,k,rn,k), and C=A+B=hmn,k(ln,k,rn,k). Indeed, the lemma implies that(3.4)D-1=A-1CB-1=hmn,k(ln,k,mn,k)-1hmn,k(ln,k,rn,k)hmn,k(ln,k,mn,k)-1=𝚺n,k-1.(i)
We have(3.5)Ln,k+Rn,k=g(mn,k)T(h(ln,k,mn,k)-1+h(mn,k,rn,k)-1)σn,k=g(mn,k)T𝚺n,k-1σn,k=g(mn,k)T(σn,k-1)T,
which ends the demonstration of the proposition.Let us defineL0,0=(h(l0,0,r0,0))-1g-1(r0,0)σ0,0. With this notations we define the functions in a compact form as follows.Definition 3.3.
For every(n,k) in ℐ with n≠0, the continuous functions ψn,k are defined on their support Sn,k as
(3.6)ψn,k(t)={g(t)h(ln,k,t)⋅Ln,k,ln,k≤t≤mn,k,g(t)h(t,rn,k)⋅Rn,k,mn,k≤t≤rn,k,
and the basis element ψ0,0 is given on [0,1] by
(3.7)ψ0,0(t)=g(t)h(l0,0,t)⋅L0,0.The definition implies thatψn,k are continuous functions in the space of piecewise derivable functions with piecewise continuous derivative which takes value zero at zero. We denote such a space by C01([0,1],ℝd×d).Before studying the property of the functionsψn,k, it is worth remembering that their definitions include the choice of a square root σn,k of Σn,k. Properly speaking, there is thus a class of bases ψn,k and all the points we develop in the sequel are valid for this class. However, for the sake of simplicity, we consider from now on that the basis under scrutiny results from choosing the unique square root σn,k that is lower triangular with positive diagonal entries (the Cholesky decomposition).


#### 3.1.1. Underlying System of Orthonormal Functions

We first introduce a family of functionsϕn,k and show that it constitutes an orthogonal basis on a certain Hilbert space. The choice of this basis can seem arbitrary at first sight, but the definition of these function will appear natural for its relationship with the functions ψn,k and Φn,k that is made explicit in the sequel, and the mathematical rigor of the argument lead us to choose this apparently artificial introduction.Definition 3.4.
For every(n,k) in ℐ  with n≠0, we define a continuous function ϕn,k:[0,1]→ℝm×d which is zero outside its support Sn,k and has the expressions:
(3.8)ϕn,k(t)={f(t)T⋅Ln,kifln,k≤t<mn,k,f(t)T⋅Rn,kifmn,k≤t<rn,k.
The basis element ϕ0,0 is defined on [0,1] by
(3.9)ϕ0,0(t)=f(t)T⋅L0,0.Remark that the definitions make apparent the fact that these two families of functions are linked for all(n,k) in ℐ through the simple relation(3.10)ψn,k′=α⋅ψn,k+Γ⋅ϕn,k.
Moreover, this collection of functions ϕn,k constitutes an orthogonal basis of functions, in the following sense.Proposition 3.5.
LetLf2 be the closure of
(3.11){u:[0,1]⟶Rm∣∃v∈L2([0,1],Rd),u=fT⋅v},
equipped with the natural norm of L2([0,1],ℝm). It is a Hilbert space, and moreover, for all 0≤j<d, the family of functions cj(ϕn,k) defined as the columns of ϕn,k, namely
(3.12)cj(ϕn,k)=[(ϕn,k)i,j]0≤i<m,
forms a complete orthonormal basis of Lf2.Proof.
The spaceLf2 is clearly a Hilbert space as a closed subspace of the larger Hilbert space L2([0,1],ℝm) is equipped with the standard scalar product:
(3.13)∀u,v∈L2([0,1],Rd),(u,v)=∫01u(t)T⋅v(t)dt.
We now proceed to demonstrate that the columns of ϕn,k form an orthonormal family which generates a dense subspace of Lf2. To this end, we define M([0,1],ℝm×d) as the space of functions
(3.14){A:[0,1]⟶Rm×d∣∀j:0≤j<d,t⟼[Ai,j(t)]0≤i<m∈L2([0,1],Rm)},
that is, the space of functions that take values in the set of m×d-matrices whose columns are in L2([0,1],ℝm). This definition allows us to define the bilinear function 𝒫:M([0,1],ℝm×d)×M([0,1],ℝm×d)→ℝd×d as
(3.15)P(A,B)=∫01A(t)T⋅B(t)dtsatisfyingP(B,A)=P(A,B)T,
and we observe that the columns of ϕn,k form an orthonormal system if and only if
(3.16)∀((p,q),(n,k))∈I×I,P(ϕn,k,ϕp,q)=∫01ϕn,k(t)T⋅ϕp,q(t)dt=δp,qn,kId,
where δp,qn,k is the Kronecker delta function, whose value is 1 if n=p and k=q, and 0 otherwise.
First of all, since the functionsϕn,k are zero outside the interval Sn,k, the matrix 𝒫(ϕn,k,ϕp,q) is nonzero only if Sn,k∩Sp,q≠∅. In such cases, assuming that n≠p and, for example, that n<p, we necessarily have Sn,k strictly included in Sp,q: more precisely, Sn,k is either included in the left-child support Sp+1,2q or in the right-child support Sp+1,2q+1 of Sp,q. In both cases, writing the matrix 𝒫(ϕn,k(t),ϕp,q) shows that it is expressed as a matrix product whose factors include 𝒫(ϕn,k,fT). We then show that
(3.17)P(ϕn,k,fT)=∫01ϕn,k(t)T⋅f(t)T=Ln,kT⋅∫ln,kmn,kf(u)⋅fT(u)du-Rn,kT⋅∫mn,krn,kf(u)⋅fT(u)du,=Ln,kT⋅h(ln,k,mn,k)-Rn,kT⋅h(mn,k,rn,k)=σn,kTg-1(mn,k)T-σn,kTg-1(mn,k)T,
which entails that 𝒫(ϕn,k,fT)=0 if n<p. If n>p, we remark that 𝒫(ϕn,k,ϕp,q)=𝒫(ϕp,q,ϕn,k)T, and we conclude that 𝒫(ϕn,k,ϕp,q)=0 from the preceding case. For  n=p, we directly compute for n>0 the only nonzero term
(3.18)P(ϕn,k,ϕn,k)=Ln,kT⋅∫ln,kmn,kf(u)⋅fT(u)du⋅Ln,k+Rn,kT⋅∫mn,krn,kf(u)⋅fT(u)du⋅Rn,k,=σn,kTg-1(mn,k)T(h(ln,k,mn,k))-1g-1(mn,k)σn,k+σn,kTg-1(mn,k)T(h(mn,k,rn,k))-1g-1(mn,k)σn,k.
Using the passage relationship between the symmetric functions h and hmn,k given in (2.7), we can then write
(3.19)P(ϕn,k,ϕn,k)=σn,kT(hmn,k(ln,k,mn,k))-1σn,k+σn,kT(hmn,k(mn,k,rn,k))-1σn,k.
Proposition 3.1 implies that hmn,k(ln,k,mn,k)-1+hmn,k(mn,k,rn,k)-1=Σn,k-1=(σn,k-1)Tσn,k-1 which directly implies that 𝒫(ϕn,k,ϕn,kT)=Id. For n=0, a computation of the exact same flavor yields that 𝒫(ϕ0,0,ϕ0,0)=Id. Hence, we have proved that the collection of columns of ϕn,k forms an orthonormal family of functions in Lf2 (the definition of ϕn,k clearly states that its columns can be written in the form of elements of Lf2).
The proof now amounts showing the density of the family of functions we consider. Before showing this density property, we introduce for all(n,k) in ℐ the functions Pn,k:[0,1]→ℝd×d with support on Sn,k defined by
(3.20)Pn,k(t)={Ln,kifln,k≤t<mn,k,-Rn,kifmn,k≤t<rn,k,n≠0,P0,0(t)=L0,0.
Showing that the family of columns of ϕn,k is dense in Lf2 is equivalent to show that the column vectors of the matrices Pn,k seen as a function of t are dense in L2([0,1],ℝd). It is enough to show that the span of such functions contains the family of piecewise continuous ℝd-valued functions that are to be constant on Sn,k, (n,k) in ℐ (the density of the endpoints of the partition ∪N∈ℕDN entails that the latter family generates L2([0,1],ℝd)).
In fact, we show that the span of functions(3.21)VN=span⁡{t⟼cj(Pn,k)(t)∣0≤j<d,(n,k)∈IN}
is exactly equal to the space KN of piecewise continuous functions from [0,1] to ℝd that are constant on the supports SN+1,k, for any (N+1,k) in ℐ. The fact that VN is included in KN is clear from the fact that the matrix-valued functions PN,k are defined constant on the support SN+1,k, for (N,k) in I.
We prove thatKN is included in VN by induction on N≤0. The property is clearly true at rank N=0 since P0,0 is then equal to the constant invertible matrix L0,0. Assuming that the proposition true at rank N-1 for a given N>0, let us consider a piecewise continuous function c:[0,1]→ℝd in KN-1. Remark that, for every (N,k) in ℐ, the function c can only take two values on SN,k and can have discontinuity jump in mN,k: let us denote these jumps as
(3.22)dN,k=c(mN,k+)-c(mN,k-).
Now, remark that for every (N,k) in ℐ, the matrix-valued functions PN,k take  only two matrix values on SN,k, namely, LN,k and -RN,k. From Proposition 3.1, we know that LN,k+RN,k=MN,k is invertible. This fact directly entails that there exist  vectors aN,k, for any (N,k) in ℐ, such that dN,k=(LN,k+RN,k)(-aN,k). We then necessarily have that the function c′=c+Pn,k·an,k is piecewise constant on the supports SN,k, (N,k) in ℐ. By recurrence hypothesis, c′ belongs to VN-1, so that c belongs to VN, and we have proved that KN⊂VN. Therefore, the space generated by the column vectors Pn,k is dense in L2[0,1], which completes the proof that the functions t↦[(ϕn,k(t))i,j]0≤i<m form a complete orthonormal family of L2[0,1].The fact that the column functions ofϕn,k form a complete orthonormal system of Lf2 directly entails the following decomposition of the identity on Lf2.Corollary 3.6.
Ifδ is the real delta Dirac function, one has
(3.23)∑(n,k)∈Iϕn,k(t)⋅ϕn,kT(s)=δ(t-s)IdLf2.Proof.
Indeed it easy to verify that, for allv in Lf2, we have for all N>0(3.24)∫U∑(n,k)∈IN(ϕn,k(t)⋅ϕn,kT(s))v(s)ds=∑(n,k)∈INϕn,k(t)⋅P(ϕn,k,v)=∑(n,k)∈IN∑p=0d-1cp(ϕn,k)(cp(ϕn,k),v),
where (cp(ϕn,k),v) denotes the inner product in Lf2 between v and the p-column of ψn,k. Therefore, by the Parseval identity, we have in the Lf2 sense
(3.25)∫U∑(n,k)∈I(ϕn,k(t)⋅ϕn,kT(s))v(s)ds=v(t).From now on, abusing language, we will say that the family ofℝm×d-valued functions ϕn,k is an orthonormal family of functions to refer to the fact that the columns of such matrices form orthonormal set of Lf2. We now make explicit the relationship between this orthonormal basis and our functions (ψn,k) derived in our analysis of the  multidimensional Gauss-Markov processes.


#### 3.1.2. Generalized Dual Operators

The Integral OperatorK
The basisϕn,k is of great interest in this paper for its relationship to the functions ψn,k that naturally arise in the decomposition of the Gauss-Markov processes. Indeed, the collection ψn,k can be generated from the orthonormal basis ϕn,k through the action of the integral operator 𝒦 defined on L2([0,1],ℝm) into L2([0,1],ℝd) by
(3.26)u↦K[u]={t⟼g(t)⋅∫U1[0,t](s)f(s)u(s)ds},
where U⊃[0,1] is an open set and, for any set E⊂U,1E(·) denotes the indicator function of E. Indeed, realizing that 𝒦 acts on M([0,1],ℝm×d) into M([0,1],ℝd×d) through
(3.27)∀A∈M([0,1],Rm×d),K[A]=[K[c0(A)],…,K[cd-1(A)]],
where cj(A) denotes the jth ℝm-valued column function of A, we easily see that for all (n,k) in ℐ, 0≤t≤1,
(3.28)ψn,k(t)=g(t)⋅∫0tf(s)⋅ϕn,k(s)ds=K[ϕn,k](t).
It is worth noticing that the introduction of the operator 𝒦 can be considered natural since it characterizes the centered Gauss-Markov process X through loosely writing X=𝒦[dW].
In order to exhibit a dual family of functions to the basisψn,k, we further investigate the property of the integral operator 𝒦. In particular, we study the existence of an inverse operator 𝒟, whose action on the orthonormal basis ϕn,k will conveniently provide us with a dual basis to ψn,k. Such an operator does not always exist; nevertheless, under special assumptions, it can be straightforwardly expressed as a generalized differential operator.The Differential OperatorD
Here, we make the assumptions thatm=d, that, for all t, f(t) is invertible in ℝd×d, and that f and f-1 have continuous derivatives, which especially implies that Lf2=L2(ℝd). In this setting, we define the space D0(U,ℝd) of functions in C0∞(U,ℝd) that are zero at zero  and denote by D0′(U,ℝd) its dual in the space of distributions (or generalized functions). Under the assumptions just made, the operator 𝒦:D0(U,ℝd)↦D0(U,ℝd) admits the differential operator 𝒟:D0(U,ℝd)↦D0(U,ℝd) defined by
(3.29)u∈D0(U,Rd)⟼D[u]={t⟼f-1(t)ddt(g-1(t)u(t))}
as its inverse, that is, when restricted to D0(U,ℝd), we have 𝒟∘𝒦=𝒦∘𝒟=Id on D0(U,ℝd). The dual operators of 𝒦 and 𝒟 are expressed, for any u in D0(U,ℝd), as
(3.30)D*[u]={t⟼-(g-1(t))Tddt((f-1(t))Tu(t))},K*[u]={t⟼-f(t)T∫U1[0,t](s)gT(s)u(s)ds}.
They satisfy (from the properties of 𝒦 and 𝒟) 𝒟*∘𝒦*=𝒦*∘𝒟*=Id on D0(U,ℝd). By dual pairing, we extend the definition of the operators 𝒦, 𝒟 as well as their dual operators, to the space of generalized function D0′(U,ℝd). In details, for any distribution T in D0′(U,ℝd) and test function u in D0(U,ℝd), define 𝒦 and 𝒦* by
(3.31)(D[T],u)=(T,D*[u]),(K[T],u)=(T,K*[u]),
and reciprocally for the dual operators 𝒟* and 𝒦*.Candidate Dual Basis
We are now in a position to use the orthonormality ofϕn,k to infer a dual family of the basis ψn,k. For any function u in L2(U,ℝd), the generalized function 𝒦[u] belongs to C0(U,ℝd), the space of continuous functions that are zero at zero. We equip this space with the uniform norm and denote its topological dual R0(U,ℝd), the set of d-dimensional Radon measures with R0(U,ℝd)⊂D0′(U,ℝd). Consequently, operating in the Gelfand triple
(3.32)C0(U,Rd)⊂L2(U,Rd)⊂R0(U,Rd),
we can write, for any function u, v in L2(U,ℝd)⊂R0(U,ℝd),
(3.33)(u,v)=((D∘K)[u],v)=(K[u],D*[v]).
The first equality stems from the fact that, when 𝒦 and 𝒟 are seen as generalized functions, they are still inverse of each other, so that in particular 𝒟∘𝒦=Id on L2(U,ℝd). The dual pairing associated with the Gelfand triple (3.32) entails the second equality where 𝒟* is the generalized operator defined on D0′(U,ℝd) and where 𝒟*[v] is in R0(U,ℝd).
As a consequence, defining the functionsδn,k in R0(U,ℝd×d), the d×d-dimensional space of Radon measures, by
(3.34)δn,k=D*(ϕn,k)=[D*[ci(ϕn,k)],…,D*[cj(ϕn,k)]]
provides us with a family of d×d-generalized functions which are dual to the family ψn,k in the sense that, for all ((n,k),(p,q)) in ℐ×ℐ, we have
(3.35)P(δn,k,ψp,q)=δp,qn,kId,
where the definition of 𝒫 has been extended through dual pairing: given any A in R0(U,ℝm×d) and any B in C0(U,ℝm×d), we have
(3.36)P(A,B)=[(ci(A),cj(B))]0≤i,j<d
with (ci(A),cj(B)) denoting the dual pairing between the ith column of A taking value in R0(U,ℝd) and the jth column of B taking value in C0(U,ℝd). Under the favorable hypothesis of this section, the d×d-generalized functions δn,k can actually be easily computed  since considering the definition of ϕn,k shows that the functions (f-1)T·ϕn,k have support Sn,k and are constant on Sn+1,2k and Sn+1,2k+1 in ℝd×d. Only the discontinuous jumps in ln,k, mn,k, and rn,k intervene, leading to expressing for (n,k) in ℐ, n≠0(3.37)δn,k(t)=(g(t)-1)T⋅(Mn,kδ(t-mn,k)-(Ln,kδ(t-ln,k)+Rn,kδ(t-rn,k)))
and δ0,0(t)=(g(t)-1)T·L0,0, where δ(·) denotes the standard delta Dirac function (centered in 0). These functions can be extended to the general setting of the paper   since its expressions do not involve the assumptions made on the invertibility and smoothness of f(t). We now show that these functions, when defined in the general setting, still provide a dual basis of the functions ψn,k.


#### 3.1.3. Dual Basis of Generalized Functions

The expression of the basisδn,k that has been found under favorable assumptions makes no explicit reference to these assumptions. It suggests   defining functions δn,k formally as linear combination of Dirac functions acting by duality on C0(U,ℝd×d).Definition 3.7.
For(n,k) in ℐ, the family of generalized functions δn,k in R0(U,ℝd×d) is given  by (n≠0)
(3.38)δn,k(t)=(g(t)-1)T⋅(Mn,kδ(t-mn,k)-(Ln,kδ(t-ln,k)+Rn,kδ(t-rn,k))),
and δ0,0(t)=(g(t)-1)T·L0,0, where δ is the standard Dirac distribution.Notice that the basisδn,k is defined for the open set U. For the sake of consistency, we extend the definition of the families ψn,k and ϕn,k on U by setting them to zero on U∖[0,1], except for ψ0,0, which is continued for t>1 by a continuous function c that is compactly supported in [1,a) for a given a in U, a>1 and satisfies c(1)=ψ0,0(1).We can now formulate the following.Proposition 3.8.
Given the dual pairing inC0(U)⊂L2(U)⊂R(U) where U is a bounded open set of ℝ containing [0,1], the family of continuous functions ψn,k in C0(U) admits, for dual family in R(U), the set of distributions δn,k.Proof.
We have to demonstrate that, for all((n,k),(p,q)) in ℐ×ℐ,
(3.39)P(δp,q,ψn,k)=δp,qn,kId.
Suppose first that n,p>0. If p<n, 𝒫(δn,k,ψp,q) can only be nonzero if the support Sp,q is strictly included in Sn,k. We then have
(3.40)P(δp,q,ψn,k)=Mp,qTg-1(mp,q)ψn,k(mp,q)-(Lp,qTg-1(lp,q)ψn,k(lp,q)+Rp,qTg-1(rp,q)ψn,k(rp,q)).
Assume that Sp,q is to the left of mn,k, that is, Sp,q is a left child of Sn,k in the nested binary tree of supports and write
(3.41)P(δp,q,ψn,k)=(Mp,qT(h(ln,k,mp,q)-Lp,qTh(ln,k,lp,q))-Rp,qTh(ln,k,rp,q))Ln,k.
Using the fact that Mp,q=Lp,q+Rp,q and that the function h(x,y), as any integral between x and y, satisfies the chain rule h(x,y)=h(x,z)+h(z,y) for all (x,y,z), we obtain
(3.42)P(δp,q,ψn,k)=(-Lp,qT(h(ln,k,mp,q)-h(ln,k,lp,q))+Rp,qT(h(ln,k,mp,q)-h(ln,k,rp,q)))Ln,k=(-Lp,qTh(lp,q,mp,q)+Rp,qTh(rp,q,mp,q))Ln,k=(-σp,qT(g-1(mp,q))T(h(lp,q,mp,q)-1)T⋅h(lp,q,mp,q)+σp,qT(g-1(mp,q))T(h(mp,q,rp,q)-1)T⋅h(mp,q,rp,q))⋅Ln,k=0.
The same result is true if Sp,q is a right child of Sn,k in the nested binary tree of supports. If p=n, necessarily the only nonzero term is for q=p, that is,
(3.43)P(δp,q,ψn,k)=Mn,kTg-1(mn,k)ψ(mn,k)=Mn,kTh(ln,k,mn,k)Ln,k=σp,q-1g(mn,k)h(ln,k,mn,k)h(ln,k,mn,k)-1g-1(mn,k)σp,q=Id.
If p>n, 𝒫(δn,k,ψp,q) can only be nonzero if the support Sn,k is included in Sp,q, but then ψn,k is zero in lp,q, mp,q, rp,q so that 𝒫(δn,k,ψp,q)=0.
Otherwise, ifn=0 and p>0, we directly have
(3.44)P(δp,q,ψ0,0)=Mp,qTg-1(mp,q)ψ0,0(mp,q)-(Lp,qTg-1(lp,q)ψ0,0(lp,q)+Rp,qTg-1(rp,q)ψ0,0(rp,q)),=(-Lp,qTh(lp,q,mp,q)+Rp,qTh(mp,q,rp,q))L0,0,=(-σp,qTg-1(mp,q)T+σp,qTg-1(mp,q)T)Ln,k,=0.
Finally, if p=0, given the simple form of δ0,0 with a single Dirac function centered in r0,0, we clearly have 𝒫(δ0,0,ψn,k)=0, and if n>0(3.45)P(δ0,0,ψ0,0),=L0,0Th(l0,0,r0,0)L0,0,=σ0,0T(g-1(r0,0))TL0,0,=σ0,0T(g-1(r0,0))T(h(l0,0,r0,0)L0,0)-1g-1(r0,0)σ0,0,=σ0,0TCr0,0-1σ0,0,
and using the fact that (by definition of σ0,0) we have σ0,0·σ0,0T=Cr0,0, this last expression is equal to
(3.46)P(δ0,0,ψ0,0)=σ0,0T(σ0,0T)-1⋅(σ0,0)-1σ0,0=Id,
which completes the proof.This proposition directly implies the main result of the section.Theorem 3.9.
The collection of functions(ψn,k;(n,k)∈I) constitutes a Schauder basis of functions on C0(U,ℝd), that is, any element of C0(U,ℝd) can be written in a unique way as a sum of coefficients an,k multiplied by ψn,k.This theorem provides us with a complementary view of stochastic processes: in addition to the standard sample paths view, this structure allows to see the Gauss-Markov processes as coefficients on the computed basis. This duality is developed in the sequel.


### 3.2. The Sample Paths Space

#### 3.2.1. The Construction Application

The Schauder basis of functions with compact supports constructed allows to define functions by considering the coefficients on this basis, which constitute sequences of real numbers in the space(3.47)ξΩ={ξ={ξn,k}I;∀(n,k)∈I,ξn,k∈Rd}=(Rd)I.
We equip ξΩ with the uniform norm ∥ξ∥∞=sup(n,k)∈ℐ|ξn,k|, where we write |ξn,k|=sup0≤i<d|(ξn,k)i|. We denote by ℬ(ξΩ) the Borelian sets of the topology induced by the uniform norm and we recall that C(ξΩ), the cylinder sets of ξΩ, form a generative family of Borelian sets. Remark that not any sequence of coefficients provides a continuous function, and one needs to assume a certain decrease in the coefficients to get convergence. A sufficient condition to obtain convergent sequences is to consider coefficients in the space(3.48)ξΩ′={ξ∈ξΩ∣∃δ∈(0,1),∃N∈N,∀(n,k)∈I∖IN,|ξn,k|<2nδ/2}.
This set is clearly a Borelian set of ξΩ  since it can be written as a countable intersection and union of cylinder, namely, by denoting by 𝒥 the set of finite subset of ℕ and δp=1-1/p, p>1,(3.49)ξΩ′=⋃p>1⋃J∈J⋂n∈N∖J{ξ∈ξΩ∣max⁡0≤k<2n-1|ξn,k|<2nδp/2}.
It is also easy to verify that it forms a vectorial subspace of ξΩ.After these definitions, we are in position to introduce the following useful function.Definition 3.10.
One denotes byΨN the partial construction application:
(3.50)ΨN={ξΩ,⟶C0([0,1],Rd),ξ,⟼∑(n,k)∈INψn,k(t)⋅ξn,k,
where the C0([0,1],ℝd) is the d-dimensional Wiener space, which is complete under the uniform norm ∥x∥∞=sup0≤t≤1|x(t)|.This sequence of partial construction applications is shown to converge to the construction application in the following.Proposition 3.11.
For everyξ in ξΩ′, ΨN(ξ) converges uniformly toward a continuous function in C0([0,1],ℝd). One will denote this function Ψ(ξ), defined as
(3.51)Ψ:{ξΩ′⟶C0([0,1],Rd),ξ⟼∑(n,k)∈Iψn,k(t)⋅ξn,k,
and this application will be referred to as the construction application.This proposition is proved in AppendixD. The image of this function constitutes a subset of the Wiener space continuous functions C0([0,1],ℝd). Let us now define the vectorial subspace xΩ′=Ψ(ξΩ′) of C0([0,1],ℝd) so that Ψ appears as a bijection.It is important to realize that, in the multidimensional case, the spacexΩ′ depends on Γ and α in a nontrivial way. For instance, assuming that α=0, the space xΩ′ depends obviously crucially on the rank of Γ. To fix the idea, for a given constant Γ(t)=[0,0⋯1]T in ℝd×1, we expect the space xΩ′ to only include sample paths of C0([0,1],ℝd) for which the n-1 first components are constant. Obviously, a process with such sample paths is degenerated in the sense that its covariance matrix is not invertible.Yet, if we additionally relax the hypothesis thatα≠0, the space xΩ′ can be dramatically altered: if we take(3.52)α(t)=[01⋱⋱⋱10],
the space xΩ′ will represent the sample space of the d-1-integrated Wiener process, a nondegenerate d-dimensional process we fully develop in the example section.However, the situation is much simpler in the one-dimensional case: because the uniform convergence of the sample paths is preserved as long asα is continuous and Γ is nonzero through (D.8),  the definition xΩ′ does not depend on α or Γ. Moreover, in this case, the space xΩ′ is large enough to contain reasonably regular functions as proved in Appendix D, Proposition 3.In the case of thed-1-integrated Wiener process, the space xΩ′ clearly contains the functions {f=(fd-1,…,f0)∣f0∈H,fi′=fi-1,0<i<d}.This remark does not hold  that the spacexΩ′ does not depend on α as long as α is continuous  because the uniform convergence of the sample paths is preserved through the change of basis of expansion ψn,k through (D.8).We equip the spacexΩ′ with the topology induced by the uniform norm on C0([0,1],ℝd). As usual, we denote ℬ(xΩ′) the corresponding Borelian sets. We prove in Appendix D the following.Proposition 3.12.
The functionΨ:(ξΩ′,ℬ(ξΩ′))→(xΩ′,ℬ(xΩ′)) is a bounded continuous bijection.We therefore conclude that we dispose of a continuous bijection mapping the coefficients onto the sample paths,Ψ. We now turn to study its inverse, the coefficient application, mapping sample paths on coefficients over the Schauder basis.


#### 3.2.2. The Coefficient Application

In this section, we introduce and study the properties of the following function.Definition 3.13.
One callscoefficient application and denotes by Ξ the function defined by
(3.53)Ξ:{C0([0,1],Rd)⟶Ωξ=(Rd)I,x⟼Δ(x)={Δ(x)}(n,k)∈Iwith{Δ(x)}n,k=P(δn,k,x).Should a functionx admit a uniformly convergent decomposition in terms of the basis of elements ψn,k, the function Δ gives its coefficients in such a representation. More precisely, we have the following.Theorem 3.14.
The functionΔ:(xΩ′,ℬ(xΩ′))→(ξΩ′,ℬ(ξΩ′)) is a measurable linear bijection whose inverse is Ψ=Δ-1.The proof of this theorem is provided in AppendixD.


## 3.1. System of Dual Bases

To complete this program, we need to introduce some quantities that will play a crucial role in expressing the familyψn,k as a Schauder basis for some given space. In   (2.39), two constant matrices ℝd×d appear  that will have a particular importance in the sequel for (n,k) in ℐ with n≠0:(3.2)Ln,k=gT(mn,k)(hmn,k(ln,k,mn,k))-1σn,k=(h(ln,k,mn,k))-1g-1(mn,k)σn,k,Rn,k=gT(mn,k)(hmn,k(mn,k,rn,k))-1σn,k=(h(mn,k,rn,k))-1g-1(mn,k)σn,k,
where h stands for h0. We further define the matrix(3.3)Mn,k=gT(mn,k)σn,k-1T
and we recall that σn,k is a square root of Σn,k, the covariance matrix of Xmn,k, conditionally to Xln,k and Xrn,k, given in (2.29). We stress that the matrices Ln,k,Rn,k, Mn,k, and Σn,k are all invertible and satisfy the important following properties.Proposition 3.1.
For all(n,k) in ℐ, n≠0, one has:(i)
Mn,k=Ln,k+Rn,k(ii)
Σn,k-1=(hmn,k(ln,k,mn,k))-1+(hmn,k(mn,k,rn,k))-1.To prove this proposition, we first establish the following simple lemma of linear algebra.Lemma 3.2.
Given two invertible matricesA and B in GLn(ℝ) such that C=A+B is also invertible, if one defines D=AC-1B, one has the following properties:(i)
D=AC-1B=BC-1A(ii)
D-1=A-1+B-1.Proof.
(i)
D=AC-1B=(C-B)C-1B=B-BC-1B=B(I-C-1B)=BC-1(C-B)=BC-1A.(ii)
(A-1+B-1)D=A-1D+B-1D=A-1AC-1B+B-1BC-1A=C-1(B+A)=C-1C=I.Proof of Proposition3.1.
(ii)
Directly stems from Lemma3.2, item (ii) by posing A=hmn,k(ln,k,mn,k), B=hmn,k(mn,k,rn,k), and C=A+B=hmn,k(ln,k,rn,k). Indeed, the lemma implies that(3.4)D-1=A-1CB-1=hmn,k(ln,k,mn,k)-1hmn,k(ln,k,rn,k)hmn,k(ln,k,mn,k)-1=𝚺n,k-1.(i)
We have(3.5)Ln,k+Rn,k=g(mn,k)T(h(ln,k,mn,k)-1+h(mn,k,rn,k)-1)σn,k=g(mn,k)T𝚺n,k-1σn,k=g(mn,k)T(σn,k-1)T,
which ends the demonstration of the proposition.Let us defineL0,0=(h(l0,0,r0,0))-1g-1(r0,0)σ0,0. With this notations we define the functions in a compact form as follows.Definition 3.3.
For every(n,k) in ℐ with n≠0, the continuous functions ψn,k are defined on their support Sn,k as
(3.6)ψn,k(t)={g(t)h(ln,k,t)⋅Ln,k,ln,k≤t≤mn,k,g(t)h(t,rn,k)⋅Rn,k,mn,k≤t≤rn,k,
and the basis element ψ0,0 is given on [0,1] by
(3.7)ψ0,0(t)=g(t)h(l0,0,t)⋅L0,0.The definition implies thatψn,k are continuous functions in the space of piecewise derivable functions with piecewise continuous derivative which takes value zero at zero. We denote such a space by C01([0,1],ℝd×d).Before studying the property of the functionsψn,k, it is worth remembering that their definitions include the choice of a square root σn,k of Σn,k. Properly speaking, there is thus a class of bases ψn,k and all the points we develop in the sequel are valid for this class. However, for the sake of simplicity, we consider from now on that the basis under scrutiny results from choosing the unique square root σn,k that is lower triangular with positive diagonal entries (the Cholesky decomposition).


### 3.1.1. Underlying System of Orthonormal Functions

We first introduce a family of functionsϕn,k and show that it constitutes an orthogonal basis on a certain Hilbert space. The choice of this basis can seem arbitrary at first sight, but the definition of these function will appear natural for its relationship with the functions ψn,k and Φn,k that is made explicit in the sequel, and the mathematical rigor of the argument lead us to choose this apparently artificial introduction.Definition 3.4.
For every(n,k) in ℐ  with n≠0, we define a continuous function ϕn,k:[0,1]→ℝm×d which is zero outside its support Sn,k and has the expressions:
(3.8)ϕn,k(t)={f(t)T⋅Ln,kifln,k≤t<mn,k,f(t)T⋅Rn,kifmn,k≤t<rn,k.
The basis element ϕ0,0 is defined on [0,1] by
(3.9)ϕ0,0(t)=f(t)T⋅L0,0.Remark that the definitions make apparent the fact that these two families of functions are linked for all(n,k) in ℐ through the simple relation(3.10)ψn,k′=α⋅ψn,k+Γ⋅ϕn,k.
Moreover, this collection of functions ϕn,k constitutes an orthogonal basis of functions, in the following sense.Proposition 3.5.
LetLf2 be the closure of
(3.11){u:[0,1]⟶Rm∣∃v∈L2([0,1],Rd),u=fT⋅v},
equipped with the natural norm of L2([0,1],ℝm). It is a Hilbert space, and moreover, for all 0≤j<d, the family of functions cj(ϕn,k) defined as the columns of ϕn,k, namely
(3.12)cj(ϕn,k)=[(ϕn,k)i,j]0≤i<m,
forms a complete orthonormal basis of Lf2.Proof.
The spaceLf2 is clearly a Hilbert space as a closed subspace of the larger Hilbert space L2([0,1],ℝm) is equipped with the standard scalar product:
(3.13)∀u,v∈L2([0,1],Rd),(u,v)=∫01u(t)T⋅v(t)dt.
We now proceed to demonstrate that the columns of ϕn,k form an orthonormal family which generates a dense subspace of Lf2. To this end, we define M([0,1],ℝm×d) as the space of functions
(3.14){A:[0,1]⟶Rm×d∣∀j:0≤j<d,t⟼[Ai,j(t)]0≤i<m∈L2([0,1],Rm)},
that is, the space of functions that take values in the set of m×d-matrices whose columns are in L2([0,1],ℝm). This definition allows us to define the bilinear function 𝒫:M([0,1],ℝm×d)×M([0,1],ℝm×d)→ℝd×d as
(3.15)P(A,B)=∫01A(t)T⋅B(t)dtsatisfyingP(B,A)=P(A,B)T,
and we observe that the columns of ϕn,k form an orthonormal system if and only if
(3.16)∀((p,q),(n,k))∈I×I,P(ϕn,k,ϕp,q)=∫01ϕn,k(t)T⋅ϕp,q(t)dt=δp,qn,kId,
where δp,qn,k is the Kronecker delta function, whose value is 1 if n=p and k=q, and 0 otherwise.
First of all, since the functionsϕn,k are zero outside the interval Sn,k, the matrix 𝒫(ϕn,k,ϕp,q) is nonzero only if Sn,k∩Sp,q≠∅. In such cases, assuming that n≠p and, for example, that n<p, we necessarily have Sn,k strictly included in Sp,q: more precisely, Sn,k is either included in the left-child support Sp+1,2q or in the right-child support Sp+1,2q+1 of Sp,q. In both cases, writing the matrix 𝒫(ϕn,k(t),ϕp,q) shows that it is expressed as a matrix product whose factors include 𝒫(ϕn,k,fT). We then show that
(3.17)P(ϕn,k,fT)=∫01ϕn,k(t)T⋅f(t)T=Ln,kT⋅∫ln,kmn,kf(u)⋅fT(u)du-Rn,kT⋅∫mn,krn,kf(u)⋅fT(u)du,=Ln,kT⋅h(ln,k,mn,k)-Rn,kT⋅h(mn,k,rn,k)=σn,kTg-1(mn,k)T-σn,kTg-1(mn,k)T,
which entails that 𝒫(ϕn,k,fT)=0 if n<p. If n>p, we remark that 𝒫(ϕn,k,ϕp,q)=𝒫(ϕp,q,ϕn,k)T, and we conclude that 𝒫(ϕn,k,ϕp,q)=0 from the preceding case. For  n=p, we directly compute for n>0 the only nonzero term
(3.18)P(ϕn,k,ϕn,k)=Ln,kT⋅∫ln,kmn,kf(u)⋅fT(u)du⋅Ln,k+Rn,kT⋅∫mn,krn,kf(u)⋅fT(u)du⋅Rn,k,=σn,kTg-1(mn,k)T(h(ln,k,mn,k))-1g-1(mn,k)σn,k+σn,kTg-1(mn,k)T(h(mn,k,rn,k))-1g-1(mn,k)σn,k.
Using the passage relationship between the symmetric functions h and hmn,k given in (2.7), we can then write
(3.19)P(ϕn,k,ϕn,k)=σn,kT(hmn,k(ln,k,mn,k))-1σn,k+σn,kT(hmn,k(mn,k,rn,k))-1σn,k.
Proposition 3.1 implies that hmn,k(ln,k,mn,k)-1+hmn,k(mn,k,rn,k)-1=Σn,k-1=(σn,k-1)Tσn,k-1 which directly implies that 𝒫(ϕn,k,ϕn,kT)=Id. For n=0, a computation of the exact same flavor yields that 𝒫(ϕ0,0,ϕ0,0)=Id. Hence, we have proved that the collection of columns of ϕn,k forms an orthonormal family of functions in Lf2 (the definition of ϕn,k clearly states that its columns can be written in the form of elements of Lf2).
The proof now amounts showing the density of the family of functions we consider. Before showing this density property, we introduce for all(n,k) in ℐ the functions Pn,k:[0,1]→ℝd×d with support on Sn,k defined by
(3.20)Pn,k(t)={Ln,kifln,k≤t<mn,k,-Rn,kifmn,k≤t<rn,k,n≠0,P0,0(t)=L0,0.
Showing that the family of columns of ϕn,k is dense in Lf2 is equivalent to show that the column vectors of the matrices Pn,k seen as a function of t are dense in L2([0,1],ℝd). It is enough to show that the span of such functions contains the family of piecewise continuous ℝd-valued functions that are to be constant on Sn,k, (n,k) in ℐ (the density of the endpoints of the partition ∪N∈ℕDN entails that the latter family generates L2([0,1],ℝd)).
In fact, we show that the span of functions(3.21)VN=span⁡{t⟼cj(Pn,k)(t)∣0≤j<d,(n,k)∈IN}
is exactly equal to the space KN of piecewise continuous functions from [0,1] to ℝd that are constant on the supports SN+1,k, for any (N+1,k) in ℐ. The fact that VN is included in KN is clear from the fact that the matrix-valued functions PN,k are defined constant on the support SN+1,k, for (N,k) in I.
We prove thatKN is included in VN by induction on N≤0. The property is clearly true at rank N=0 since P0,0 is then equal to the constant invertible matrix L0,0. Assuming that the proposition true at rank N-1 for a given N>0, let us consider a piecewise continuous function c:[0,1]→ℝd in KN-1. Remark that, for every (N,k) in ℐ, the function c can only take two values on SN,k and can have discontinuity jump in mN,k: let us denote these jumps as
(3.22)dN,k=c(mN,k+)-c(mN,k-).
Now, remark that for every (N,k) in ℐ, the matrix-valued functions PN,k take  only two matrix values on SN,k, namely, LN,k and -RN,k. From Proposition 3.1, we know that LN,k+RN,k=MN,k is invertible. This fact directly entails that there exist  vectors aN,k, for any (N,k) in ℐ, such that dN,k=(LN,k+RN,k)(-aN,k). We then necessarily have that the function c′=c+Pn,k·an,k is piecewise constant on the supports SN,k, (N,k) in ℐ. By recurrence hypothesis, c′ belongs to VN-1, so that c belongs to VN, and we have proved that KN⊂VN. Therefore, the space generated by the column vectors Pn,k is dense in L2[0,1], which completes the proof that the functions t↦[(ϕn,k(t))i,j]0≤i<m form a complete orthonormal family of L2[0,1].The fact that the column functions ofϕn,k form a complete orthonormal system of Lf2 directly entails the following decomposition of the identity on Lf2.Corollary 3.6.
Ifδ is the real delta Dirac function, one has
(3.23)∑(n,k)∈Iϕn,k(t)⋅ϕn,kT(s)=δ(t-s)IdLf2.Proof.
Indeed it easy to verify that, for allv in Lf2, we have for all N>0(3.24)∫U∑(n,k)∈IN(ϕn,k(t)⋅ϕn,kT(s))v(s)ds=∑(n,k)∈INϕn,k(t)⋅P(ϕn,k,v)=∑(n,k)∈IN∑p=0d-1cp(ϕn,k)(cp(ϕn,k),v),
where (cp(ϕn,k),v) denotes the inner product in Lf2 between v and the p-column of ψn,k. Therefore, by the Parseval identity, we have in the Lf2 sense
(3.25)∫U∑(n,k)∈I(ϕn,k(t)⋅ϕn,kT(s))v(s)ds=v(t).From now on, abusing language, we will say that the family ofℝm×d-valued functions ϕn,k is an orthonormal family of functions to refer to the fact that the columns of such matrices form orthonormal set of Lf2. We now make explicit the relationship between this orthonormal basis and our functions (ψn,k) derived in our analysis of the  multidimensional Gauss-Markov processes.


### 3.1.2. Generalized Dual Operators

The Integral OperatorK
The basisϕn,k is of great interest in this paper for its relationship to the functions ψn,k that naturally arise in the decomposition of the Gauss-Markov processes. Indeed, the collection ψn,k can be generated from the orthonormal basis ϕn,k through the action of the integral operator 𝒦 defined on L2([0,1],ℝm) into L2([0,1],ℝd) by
(3.26)u↦K[u]={t⟼g(t)⋅∫U1[0,t](s)f(s)u(s)ds},
where U⊃[0,1] is an open set and, for any set E⊂U,1E(·) denotes the indicator function of E. Indeed, realizing that 𝒦 acts on M([0,1],ℝm×d) into M([0,1],ℝd×d) through
(3.27)∀A∈M([0,1],Rm×d),K[A]=[K[c0(A)],…,K[cd-1(A)]],
where cj(A) denotes the jth ℝm-valued column function of A, we easily see that for all (n,k) in ℐ, 0≤t≤1,
(3.28)ψn,k(t)=g(t)⋅∫0tf(s)⋅ϕn,k(s)ds=K[ϕn,k](t).
It is worth noticing that the introduction of the operator 𝒦 can be considered natural since it characterizes the centered Gauss-Markov process X through loosely writing X=𝒦[dW].
In order to exhibit a dual family of functions to the basisψn,k, we further investigate the property of the integral operator 𝒦. In particular, we study the existence of an inverse operator 𝒟, whose action on the orthonormal basis ϕn,k will conveniently provide us with a dual basis to ψn,k. Such an operator does not always exist; nevertheless, under special assumptions, it can be straightforwardly expressed as a generalized differential operator.The Differential OperatorD
Here, we make the assumptions thatm=d, that, for all t, f(t) is invertible in ℝd×d, and that f and f-1 have continuous derivatives, which especially implies that Lf2=L2(ℝd). In this setting, we define the space D0(U,ℝd) of functions in C0∞(U,ℝd) that are zero at zero  and denote by D0′(U,ℝd) its dual in the space of distributions (or generalized functions). Under the assumptions just made, the operator 𝒦:D0(U,ℝd)↦D0(U,ℝd) admits the differential operator 𝒟:D0(U,ℝd)↦D0(U,ℝd) defined by
(3.29)u∈D0(U,Rd)⟼D[u]={t⟼f-1(t)ddt(g-1(t)u(t))}
as its inverse, that is, when restricted to D0(U,ℝd), we have 𝒟∘𝒦=𝒦∘𝒟=Id on D0(U,ℝd). The dual operators of 𝒦 and 𝒟 are expressed, for any u in D0(U,ℝd), as
(3.30)D*[u]={t⟼-(g-1(t))Tddt((f-1(t))Tu(t))},K*[u]={t⟼-f(t)T∫U1[0,t](s)gT(s)u(s)ds}.
They satisfy (from the properties of 𝒦 and 𝒟) 𝒟*∘𝒦*=𝒦*∘𝒟*=Id on D0(U,ℝd). By dual pairing, we extend the definition of the operators 𝒦, 𝒟 as well as their dual operators, to the space of generalized function D0′(U,ℝd). In details, for any distribution T in D0′(U,ℝd) and test function u in D0(U,ℝd), define 𝒦 and 𝒦* by
(3.31)(D[T],u)=(T,D*[u]),(K[T],u)=(T,K*[u]),
and reciprocally for the dual operators 𝒟* and 𝒦*.Candidate Dual Basis
We are now in a position to use the orthonormality ofϕn,k to infer a dual family of the basis ψn,k. For any function u in L2(U,ℝd), the generalized function 𝒦[u] belongs to C0(U,ℝd), the space of continuous functions that are zero at zero. We equip this space with the uniform norm and denote its topological dual R0(U,ℝd), the set of d-dimensional Radon measures with R0(U,ℝd)⊂D0′(U,ℝd). Consequently, operating in the Gelfand triple
(3.32)C0(U,Rd)⊂L2(U,Rd)⊂R0(U,Rd),
we can write, for any function u, v in L2(U,ℝd)⊂R0(U,ℝd),
(3.33)(u,v)=((D∘K)[u],v)=(K[u],D*[v]).
The first equality stems from the fact that, when 𝒦 and 𝒟 are seen as generalized functions, they are still inverse of each other, so that in particular 𝒟∘𝒦=Id on L2(U,ℝd). The dual pairing associated with the Gelfand triple (3.32) entails the second equality where 𝒟* is the generalized operator defined on D0′(U,ℝd) and where 𝒟*[v] is in R0(U,ℝd).
As a consequence, defining the functionsδn,k in R0(U,ℝd×d), the d×d-dimensional space of Radon measures, by
(3.34)δn,k=D*(ϕn,k)=[D*[ci(ϕn,k)],…,D*[cj(ϕn,k)]]
provides us with a family of d×d-generalized functions which are dual to the family ψn,k in the sense that, for all ((n,k),(p,q)) in ℐ×ℐ, we have
(3.35)P(δn,k,ψp,q)=δp,qn,kId,
where the definition of 𝒫 has been extended through dual pairing: given any A in R0(U,ℝm×d) and any B in C0(U,ℝm×d), we have
(3.36)P(A,B)=[(ci(A),cj(B))]0≤i,j<d
with (ci(A),cj(B)) denoting the dual pairing between the ith column of A taking value in R0(U,ℝd) and the jth column of B taking value in C0(U,ℝd). Under the favorable hypothesis of this section, the d×d-generalized functions δn,k can actually be easily computed  since considering the definition of ϕn,k shows that the functions (f-1)T·ϕn,k have support Sn,k and are constant on Sn+1,2k and Sn+1,2k+1 in ℝd×d. Only the discontinuous jumps in ln,k, mn,k, and rn,k intervene, leading to expressing for (n,k) in ℐ, n≠0(3.37)δn,k(t)=(g(t)-1)T⋅(Mn,kδ(t-mn,k)-(Ln,kδ(t-ln,k)+Rn,kδ(t-rn,k)))
and δ0,0(t)=(g(t)-1)T·L0,0, where δ(·) denotes the standard delta Dirac function (centered in 0). These functions can be extended to the general setting of the paper   since its expressions do not involve the assumptions made on the invertibility and smoothness of f(t). We now show that these functions, when defined in the general setting, still provide a dual basis of the functions ψn,k.


### 3.1.3. Dual Basis of Generalized Functions

The expression of the basisδn,k that has been found under favorable assumptions makes no explicit reference to these assumptions. It suggests   defining functions δn,k formally as linear combination of Dirac functions acting by duality on C0(U,ℝd×d).Definition 3.7.
For(n,k) in ℐ, the family of generalized functions δn,k in R0(U,ℝd×d) is given  by (n≠0)
(3.38)δn,k(t)=(g(t)-1)T⋅(Mn,kδ(t-mn,k)-(Ln,kδ(t-ln,k)+Rn,kδ(t-rn,k))),
and δ0,0(t)=(g(t)-1)T·L0,0, where δ is the standard Dirac distribution.Notice that the basisδn,k is defined for the open set U. For the sake of consistency, we extend the definition of the families ψn,k and ϕn,k on U by setting them to zero on U∖[0,1], except for ψ0,0, which is continued for t>1 by a continuous function c that is compactly supported in [1,a) for a given a in U, a>1 and satisfies c(1)=ψ0,0(1).We can now formulate the following.Proposition 3.8.
Given the dual pairing inC0(U)⊂L2(U)⊂R(U) where U is a bounded open set of ℝ containing [0,1], the family of continuous functions ψn,k in C0(U) admits, for dual family in R(U), the set of distributions δn,k.Proof.
We have to demonstrate that, for all((n,k),(p,q)) in ℐ×ℐ,
(3.39)P(δp,q,ψn,k)=δp,qn,kId.
Suppose first that n,p>0. If p<n, 𝒫(δn,k,ψp,q) can only be nonzero if the support Sp,q is strictly included in Sn,k. We then have
(3.40)P(δp,q,ψn,k)=Mp,qTg-1(mp,q)ψn,k(mp,q)-(Lp,qTg-1(lp,q)ψn,k(lp,q)+Rp,qTg-1(rp,q)ψn,k(rp,q)).
Assume that Sp,q is to the left of mn,k, that is, Sp,q is a left child of Sn,k in the nested binary tree of supports and write
(3.41)P(δp,q,ψn,k)=(Mp,qT(h(ln,k,mp,q)-Lp,qTh(ln,k,lp,q))-Rp,qTh(ln,k,rp,q))Ln,k.
Using the fact that Mp,q=Lp,q+Rp,q and that the function h(x,y), as any integral between x and y, satisfies the chain rule h(x,y)=h(x,z)+h(z,y) for all (x,y,z), we obtain
(3.42)P(δp,q,ψn,k)=(-Lp,qT(h(ln,k,mp,q)-h(ln,k,lp,q))+Rp,qT(h(ln,k,mp,q)-h(ln,k,rp,q)))Ln,k=(-Lp,qTh(lp,q,mp,q)+Rp,qTh(rp,q,mp,q))Ln,k=(-σp,qT(g-1(mp,q))T(h(lp,q,mp,q)-1)T⋅h(lp,q,mp,q)+σp,qT(g-1(mp,q))T(h(mp,q,rp,q)-1)T⋅h(mp,q,rp,q))⋅Ln,k=0.
The same result is true if Sp,q is a right child of Sn,k in the nested binary tree of supports. If p=n, necessarily the only nonzero term is for q=p, that is,
(3.43)P(δp,q,ψn,k)=Mn,kTg-1(mn,k)ψ(mn,k)=Mn,kTh(ln,k,mn,k)Ln,k=σp,q-1g(mn,k)h(ln,k,mn,k)h(ln,k,mn,k)-1g-1(mn,k)σp,q=Id.
If p>n, 𝒫(δn,k,ψp,q) can only be nonzero if the support Sn,k is included in Sp,q, but then ψn,k is zero in lp,q, mp,q, rp,q so that 𝒫(δn,k,ψp,q)=0.
Otherwise, ifn=0 and p>0, we directly have
(3.44)P(δp,q,ψ0,0)=Mp,qTg-1(mp,q)ψ0,0(mp,q)-(Lp,qTg-1(lp,q)ψ0,0(lp,q)+Rp,qTg-1(rp,q)ψ0,0(rp,q)),=(-Lp,qTh(lp,q,mp,q)+Rp,qTh(mp,q,rp,q))L0,0,=(-σp,qTg-1(mp,q)T+σp,qTg-1(mp,q)T)Ln,k,=0.
Finally, if p=0, given the simple form of δ0,0 with a single Dirac function centered in r0,0, we clearly have 𝒫(δ0,0,ψn,k)=0, and if n>0(3.45)P(δ0,0,ψ0,0),=L0,0Th(l0,0,r0,0)L0,0,=σ0,0T(g-1(r0,0))TL0,0,=σ0,0T(g-1(r0,0))T(h(l0,0,r0,0)L0,0)-1g-1(r0,0)σ0,0,=σ0,0TCr0,0-1σ0,0,
and using the fact that (by definition of σ0,0) we have σ0,0·σ0,0T=Cr0,0, this last expression is equal to
(3.46)P(δ0,0,ψ0,0)=σ0,0T(σ0,0T)-1⋅(σ0,0)-1σ0,0=Id,
which completes the proof.This proposition directly implies the main result of the section.Theorem 3.9.
The collection of functions(ψn,k;(n,k)∈I) constitutes a Schauder basis of functions on C0(U,ℝd), that is, any element of C0(U,ℝd) can be written in a unique way as a sum of coefficients an,k multiplied by ψn,k.This theorem provides us with a complementary view of stochastic processes: in addition to the standard sample paths view, this structure allows to see the Gauss-Markov processes as coefficients on the computed basis. This duality is developed in the sequel.


## 3.1.1. Underlying System of Orthonormal Functions

We first introduce a family of functionsϕn,k and show that it constitutes an orthogonal basis on a certain Hilbert space. The choice of this basis can seem arbitrary at first sight, but the definition of these function will appear natural for its relationship with the functions ψn,k and Φn,k that is made explicit in the sequel, and the mathematical rigor of the argument lead us to choose this apparently artificial introduction.Definition 3.4.
For every(n,k) in ℐ  with n≠0, we define a continuous function ϕn,k:[0,1]→ℝm×d which is zero outside its support Sn,k and has the expressions:
(3.8)ϕn,k(t)={f(t)T⋅Ln,kifln,k≤t<mn,k,f(t)T⋅Rn,kifmn,k≤t<rn,k.
The basis element ϕ0,0 is defined on [0,1] by
(3.9)ϕ0,0(t)=f(t)T⋅L0,0.Remark that the definitions make apparent the fact that these two families of functions are linked for all(n,k) in ℐ through the simple relation(3.10)ψn,k′=α⋅ψn,k+Γ⋅ϕn,k.
Moreover, this collection of functions ϕn,k constitutes an orthogonal basis of functions, in the following sense.Proposition 3.5.
LetLf2 be the closure of
(3.11){u:[0,1]⟶Rm∣∃v∈L2([0,1],Rd),u=fT⋅v},
equipped with the natural norm of L2([0,1],ℝm). It is a Hilbert space, and moreover, for all 0≤j<d, the family of functions cj(ϕn,k) defined as the columns of ϕn,k, namely
(3.12)cj(ϕn,k)=[(ϕn,k)i,j]0≤i<m,
forms a complete orthonormal basis of Lf2.Proof.
The spaceLf2 is clearly a Hilbert space as a closed subspace of the larger Hilbert space L2([0,1],ℝm) is equipped with the standard scalar product:
(3.13)∀u,v∈L2([0,1],Rd),(u,v)=∫01u(t)T⋅v(t)dt.
We now proceed to demonstrate that the columns of ϕn,k form an orthonormal family which generates a dense subspace of Lf2. To this end, we define M([0,1],ℝm×d) as the space of functions
(3.14){A:[0,1]⟶Rm×d∣∀j:0≤j<d,t⟼[Ai,j(t)]0≤i<m∈L2([0,1],Rm)},
that is, the space of functions that take values in the set of m×d-matrices whose columns are in L2([0,1],ℝm). This definition allows us to define the bilinear function 𝒫:M([0,1],ℝm×d)×M([0,1],ℝm×d)→ℝd×d as
(3.15)P(A,B)=∫01A(t)T⋅B(t)dtsatisfyingP(B,A)=P(A,B)T,
and we observe that the columns of ϕn,k form an orthonormal system if and only if
(3.16)∀((p,q),(n,k))∈I×I,P(ϕn,k,ϕp,q)=∫01ϕn,k(t)T⋅ϕp,q(t)dt=δp,qn,kId,
where δp,qn,k is the Kronecker delta function, whose value is 1 if n=p and k=q, and 0 otherwise.
First of all, since the functionsϕn,k are zero outside the interval Sn,k, the matrix 𝒫(ϕn,k,ϕp,q) is nonzero only if Sn,k∩Sp,q≠∅. In such cases, assuming that n≠p and, for example, that n<p, we necessarily have Sn,k strictly included in Sp,q: more precisely, Sn,k is either included in the left-child support Sp+1,2q or in the right-child support Sp+1,2q+1 of Sp,q. In both cases, writing the matrix 𝒫(ϕn,k(t),ϕp,q) shows that it is expressed as a matrix product whose factors include 𝒫(ϕn,k,fT). We then show that
(3.17)P(ϕn,k,fT)=∫01ϕn,k(t)T⋅f(t)T=Ln,kT⋅∫ln,kmn,kf(u)⋅fT(u)du-Rn,kT⋅∫mn,krn,kf(u)⋅fT(u)du,=Ln,kT⋅h(ln,k,mn,k)-Rn,kT⋅h(mn,k,rn,k)=σn,kTg-1(mn,k)T-σn,kTg-1(mn,k)T,
which entails that 𝒫(ϕn,k,fT)=0 if n<p. If n>p, we remark that 𝒫(ϕn,k,ϕp,q)=𝒫(ϕp,q,ϕn,k)T, and we conclude that 𝒫(ϕn,k,ϕp,q)=0 from the preceding case. For  n=p, we directly compute for n>0 the only nonzero term
(3.18)P(ϕn,k,ϕn,k)=Ln,kT⋅∫ln,kmn,kf(u)⋅fT(u)du⋅Ln,k+Rn,kT⋅∫mn,krn,kf(u)⋅fT(u)du⋅Rn,k,=σn,kTg-1(mn,k)T(h(ln,k,mn,k))-1g-1(mn,k)σn,k+σn,kTg-1(mn,k)T(h(mn,k,rn,k))-1g-1(mn,k)σn,k.
Using the passage relationship between the symmetric functions h and hmn,k given in (2.7), we can then write
(3.19)P(ϕn,k,ϕn,k)=σn,kT(hmn,k(ln,k,mn,k))-1σn,k+σn,kT(hmn,k(mn,k,rn,k))-1σn,k.
Proposition 3.1 implies that hmn,k(ln,k,mn,k)-1+hmn,k(mn,k,rn,k)-1=Σn,k-1=(σn,k-1)Tσn,k-1 which directly implies that 𝒫(ϕn,k,ϕn,kT)=Id. For n=0, a computation of the exact same flavor yields that 𝒫(ϕ0,0,ϕ0,0)=Id. Hence, we have proved that the collection of columns of ϕn,k forms an orthonormal family of functions in Lf2 (the definition of ϕn,k clearly states that its columns can be written in the form of elements of Lf2).
The proof now amounts showing the density of the family of functions we consider. Before showing this density property, we introduce for all(n,k) in ℐ the functions Pn,k:[0,1]→ℝd×d with support on Sn,k defined by
(3.20)Pn,k(t)={Ln,kifln,k≤t<mn,k,-Rn,kifmn,k≤t<rn,k,n≠0,P0,0(t)=L0,0.
Showing that the family of columns of ϕn,k is dense in Lf2 is equivalent to show that the column vectors of the matrices Pn,k seen as a function of t are dense in L2([0,1],ℝd). It is enough to show that the span of such functions contains the family of piecewise continuous ℝd-valued functions that are to be constant on Sn,k, (n,k) in ℐ (the density of the endpoints of the partition ∪N∈ℕDN entails that the latter family generates L2([0,1],ℝd)).
In fact, we show that the span of functions(3.21)VN=span⁡{t⟼cj(Pn,k)(t)∣0≤j<d,(n,k)∈IN}
is exactly equal to the space KN of piecewise continuous functions from [0,1] to ℝd that are constant on the supports SN+1,k, for any (N+1,k) in ℐ. The fact that VN is included in KN is clear from the fact that the matrix-valued functions PN,k are defined constant on the support SN+1,k, for (N,k) in I.
We prove thatKN is included in VN by induction on N≤0. The property is clearly true at rank N=0 since P0,0 is then equal to the constant invertible matrix L0,0. Assuming that the proposition true at rank N-1 for a given N>0, let us consider a piecewise continuous function c:[0,1]→ℝd in KN-1. Remark that, for every (N,k) in ℐ, the function c can only take two values on SN,k and can have discontinuity jump in mN,k: let us denote these jumps as
(3.22)dN,k=c(mN,k+)-c(mN,k-).
Now, remark that for every (N,k) in ℐ, the matrix-valued functions PN,k take  only two matrix values on SN,k, namely, LN,k and -RN,k. From Proposition 3.1, we know that LN,k+RN,k=MN,k is invertible. This fact directly entails that there exist  vectors aN,k, for any (N,k) in ℐ, such that dN,k=(LN,k+RN,k)(-aN,k). We then necessarily have that the function c′=c+Pn,k·an,k is piecewise constant on the supports SN,k, (N,k) in ℐ. By recurrence hypothesis, c′ belongs to VN-1, so that c belongs to VN, and we have proved that KN⊂VN. Therefore, the space generated by the column vectors Pn,k is dense in L2[0,1], which completes the proof that the functions t↦[(ϕn,k(t))i,j]0≤i<m form a complete orthonormal family of L2[0,1].The fact that the column functions ofϕn,k form a complete orthonormal system of Lf2 directly entails the following decomposition of the identity on Lf2.Corollary 3.6.
Ifδ is the real delta Dirac function, one has
(3.23)∑(n,k)∈Iϕn,k(t)⋅ϕn,kT(s)=δ(t-s)IdLf2.Proof.
Indeed it easy to verify that, for allv in Lf2, we have for all N>0(3.24)∫U∑(n,k)∈IN(ϕn,k(t)⋅ϕn,kT(s))v(s)ds=∑(n,k)∈INϕn,k(t)⋅P(ϕn,k,v)=∑(n,k)∈IN∑p=0d-1cp(ϕn,k)(cp(ϕn,k),v),
where (cp(ϕn,k),v) denotes the inner product in Lf2 between v and the p-column of ψn,k. Therefore, by the Parseval identity, we have in the Lf2 sense
(3.25)∫U∑(n,k)∈I(ϕn,k(t)⋅ϕn,kT(s))v(s)ds=v(t).From now on, abusing language, we will say that the family ofℝm×d-valued functions ϕn,k is an orthonormal family of functions to refer to the fact that the columns of such matrices form orthonormal set of Lf2. We now make explicit the relationship between this orthonormal basis and our functions (ψn,k) derived in our analysis of the  multidimensional Gauss-Markov processes.


## 3.1.2. Generalized Dual Operators

The Integral OperatorK
The basisϕn,k is of great interest in this paper for its relationship to the functions ψn,k that naturally arise in the decomposition of the Gauss-Markov processes. Indeed, the collection ψn,k can be generated from the orthonormal basis ϕn,k through the action of the integral operator 𝒦 defined on L2([0,1],ℝm) into L2([0,1],ℝd) by
(3.26)u↦K[u]={t⟼g(t)⋅∫U1[0,t](s)f(s)u(s)ds},
where U⊃[0,1] is an open set and, for any set E⊂U,1E(·) denotes the indicator function of E. Indeed, realizing that 𝒦 acts on M([0,1],ℝm×d) into M([0,1],ℝd×d) through
(3.27)∀A∈M([0,1],Rm×d),K[A]=[K[c0(A)],…,K[cd-1(A)]],
where cj(A) denotes the jth ℝm-valued column function of A, we easily see that for all (n,k) in ℐ, 0≤t≤1,
(3.28)ψn,k(t)=g(t)⋅∫0tf(s)⋅ϕn,k(s)ds=K[ϕn,k](t).
It is worth noticing that the introduction of the operator 𝒦 can be considered natural since it characterizes the centered Gauss-Markov process X through loosely writing X=𝒦[dW].
In order to exhibit a dual family of functions to the basisψn,k, we further investigate the property of the integral operator 𝒦. In particular, we study the existence of an inverse operator 𝒟, whose action on the orthonormal basis ϕn,k will conveniently provide us with a dual basis to ψn,k. Such an operator does not always exist; nevertheless, under special assumptions, it can be straightforwardly expressed as a generalized differential operator.The Differential OperatorD
Here, we make the assumptions thatm=d, that, for all t, f(t) is invertible in ℝd×d, and that f and f-1 have continuous derivatives, which especially implies that Lf2=L2(ℝd). In this setting, we define the space D0(U,ℝd) of functions in C0∞(U,ℝd) that are zero at zero  and denote by D0′(U,ℝd) its dual in the space of distributions (or generalized functions). Under the assumptions just made, the operator 𝒦:D0(U,ℝd)↦D0(U,ℝd) admits the differential operator 𝒟:D0(U,ℝd)↦D0(U,ℝd) defined by
(3.29)u∈D0(U,Rd)⟼D[u]={t⟼f-1(t)ddt(g-1(t)u(t))}
as its inverse, that is, when restricted to D0(U,ℝd), we have 𝒟∘𝒦=𝒦∘𝒟=Id on D0(U,ℝd). The dual operators of 𝒦 and 𝒟 are expressed, for any u in D0(U,ℝd), as
(3.30)D*[u]={t⟼-(g-1(t))Tddt((f-1(t))Tu(t))},K*[u]={t⟼-f(t)T∫U1[0,t](s)gT(s)u(s)ds}.
They satisfy (from the properties of 𝒦 and 𝒟) 𝒟*∘𝒦*=𝒦*∘𝒟*=Id on D0(U,ℝd). By dual pairing, we extend the definition of the operators 𝒦, 𝒟 as well as their dual operators, to the space of generalized function D0′(U,ℝd). In details, for any distribution T in D0′(U,ℝd) and test function u in D0(U,ℝd), define 𝒦 and 𝒦* by
(3.31)(D[T],u)=(T,D*[u]),(K[T],u)=(T,K*[u]),
and reciprocally for the dual operators 𝒟* and 𝒦*.Candidate Dual Basis
We are now in a position to use the orthonormality ofϕn,k to infer a dual family of the basis ψn,k. For any function u in L2(U,ℝd), the generalized function 𝒦[u] belongs to C0(U,ℝd), the space of continuous functions that are zero at zero. We equip this space with the uniform norm and denote its topological dual R0(U,ℝd), the set of d-dimensional Radon measures with R0(U,ℝd)⊂D0′(U,ℝd). Consequently, operating in the Gelfand triple
(3.32)C0(U,Rd)⊂L2(U,Rd)⊂R0(U,Rd),
we can write, for any function u, v in L2(U,ℝd)⊂R0(U,ℝd),
(3.33)(u,v)=((D∘K)[u],v)=(K[u],D*[v]).
The first equality stems from the fact that, when 𝒦 and 𝒟 are seen as generalized functions, they are still inverse of each other, so that in particular 𝒟∘𝒦=Id on L2(U,ℝd). The dual pairing associated with the Gelfand triple (3.32) entails the second equality where 𝒟* is the generalized operator defined on D0′(U,ℝd) and where 𝒟*[v] is in R0(U,ℝd).
As a consequence, defining the functionsδn,k in R0(U,ℝd×d), the d×d-dimensional space of Radon measures, by
(3.34)δn,k=D*(ϕn,k)=[D*[ci(ϕn,k)],…,D*[cj(ϕn,k)]]
provides us with a family of d×d-generalized functions which are dual to the family ψn,k in the sense that, for all ((n,k),(p,q)) in ℐ×ℐ, we have
(3.35)P(δn,k,ψp,q)=δp,qn,kId,
where the definition of 𝒫 has been extended through dual pairing: given any A in R0(U,ℝm×d) and any B in C0(U,ℝm×d), we have
(3.36)P(A,B)=[(ci(A),cj(B))]0≤i,j<d
with (ci(A),cj(B)) denoting the dual pairing between the ith column of A taking value in R0(U,ℝd) and the jth column of B taking value in C0(U,ℝd). Under the favorable hypothesis of this section, the d×d-generalized functions δn,k can actually be easily computed  since considering the definition of ϕn,k shows that the functions (f-1)T·ϕn,k have support Sn,k and are constant on Sn+1,2k and Sn+1,2k+1 in ℝd×d. Only the discontinuous jumps in ln,k, mn,k, and rn,k intervene, leading to expressing for (n,k) in ℐ, n≠0(3.37)δn,k(t)=(g(t)-1)T⋅(Mn,kδ(t-mn,k)-(Ln,kδ(t-ln,k)+Rn,kδ(t-rn,k)))
and δ0,0(t)=(g(t)-1)T·L0,0, where δ(·) denotes the standard delta Dirac function (centered in 0). These functions can be extended to the general setting of the paper   since its expressions do not involve the assumptions made on the invertibility and smoothness of f(t). We now show that these functions, when defined in the general setting, still provide a dual basis of the functions ψn,k.


## 3.1.3. Dual Basis of Generalized Functions

The expression of the basisδn,k that has been found under favorable assumptions makes no explicit reference to these assumptions. It suggests   defining functions δn,k formally as linear combination of Dirac functions acting by duality on C0(U,ℝd×d).Definition 3.7.
For(n,k) in ℐ, the family of generalized functions δn,k in R0(U,ℝd×d) is given  by (n≠0)
(3.38)δn,k(t)=(g(t)-1)T⋅(Mn,kδ(t-mn,k)-(Ln,kδ(t-ln,k)+Rn,kδ(t-rn,k))),
and δ0,0(t)=(g(t)-1)T·L0,0, where δ is the standard Dirac distribution.Notice that the basisδn,k is defined for the open set U. For the sake of consistency, we extend the definition of the families ψn,k and ϕn,k on U by setting them to zero on U∖[0,1], except for ψ0,0, which is continued for t>1 by a continuous function c that is compactly supported in [1,a) for a given a in U, a>1 and satisfies c(1)=ψ0,0(1).We can now formulate the following.Proposition 3.8.
Given the dual pairing inC0(U)⊂L2(U)⊂R(U) where U is a bounded open set of ℝ containing [0,1], the family of continuous functions ψn,k in C0(U) admits, for dual family in R(U), the set of distributions δn,k.Proof.
We have to demonstrate that, for all((n,k),(p,q)) in ℐ×ℐ,
(3.39)P(δp,q,ψn,k)=δp,qn,kId.
Suppose first that n,p>0. If p<n, 𝒫(δn,k,ψp,q) can only be nonzero if the support Sp,q is strictly included in Sn,k. We then have
(3.40)P(δp,q,ψn,k)=Mp,qTg-1(mp,q)ψn,k(mp,q)-(Lp,qTg-1(lp,q)ψn,k(lp,q)+Rp,qTg-1(rp,q)ψn,k(rp,q)).
Assume that Sp,q is to the left of mn,k, that is, Sp,q is a left child of Sn,k in the nested binary tree of supports and write
(3.41)P(δp,q,ψn,k)=(Mp,qT(h(ln,k,mp,q)-Lp,qTh(ln,k,lp,q))-Rp,qTh(ln,k,rp,q))Ln,k.
Using the fact that Mp,q=Lp,q+Rp,q and that the function h(x,y), as any integral between x and y, satisfies the chain rule h(x,y)=h(x,z)+h(z,y) for all (x,y,z), we obtain
(3.42)P(δp,q,ψn,k)=(-Lp,qT(h(ln,k,mp,q)-h(ln,k,lp,q))+Rp,qT(h(ln,k,mp,q)-h(ln,k,rp,q)))Ln,k=(-Lp,qTh(lp,q,mp,q)+Rp,qTh(rp,q,mp,q))Ln,k=(-σp,qT(g-1(mp,q))T(h(lp,q,mp,q)-1)T⋅h(lp,q,mp,q)+σp,qT(g-1(mp,q))T(h(mp,q,rp,q)-1)T⋅h(mp,q,rp,q))⋅Ln,k=0.
The same result is true if Sp,q is a right child of Sn,k in the nested binary tree of supports. If p=n, necessarily the only nonzero term is for q=p, that is,
(3.43)P(δp,q,ψn,k)=Mn,kTg-1(mn,k)ψ(mn,k)=Mn,kTh(ln,k,mn,k)Ln,k=σp,q-1g(mn,k)h(ln,k,mn,k)h(ln,k,mn,k)-1g-1(mn,k)σp,q=Id.
If p>n, 𝒫(δn,k,ψp,q) can only be nonzero if the support Sn,k is included in Sp,q, but then ψn,k is zero in lp,q, mp,q, rp,q so that 𝒫(δn,k,ψp,q)=0.
Otherwise, ifn=0 and p>0, we directly have
(3.44)P(δp,q,ψ0,0)=Mp,qTg-1(mp,q)ψ0,0(mp,q)-(Lp,qTg-1(lp,q)ψ0,0(lp,q)+Rp,qTg-1(rp,q)ψ0,0(rp,q)),=(-Lp,qTh(lp,q,mp,q)+Rp,qTh(mp,q,rp,q))L0,0,=(-σp,qTg-1(mp,q)T+σp,qTg-1(mp,q)T)Ln,k,=0.
Finally, if p=0, given the simple form of δ0,0 with a single Dirac function centered in r0,0, we clearly have 𝒫(δ0,0,ψn,k)=0, and if n>0(3.45)P(δ0,0,ψ0,0),=L0,0Th(l0,0,r0,0)L0,0,=σ0,0T(g-1(r0,0))TL0,0,=σ0,0T(g-1(r0,0))T(h(l0,0,r0,0)L0,0)-1g-1(r0,0)σ0,0,=σ0,0TCr0,0-1σ0,0,
and using the fact that (by definition of σ0,0) we have σ0,0·σ0,0T=Cr0,0, this last expression is equal to
(3.46)P(δ0,0,ψ0,0)=σ0,0T(σ0,0T)-1⋅(σ0,0)-1σ0,0=Id,
which completes the proof.This proposition directly implies the main result of the section.Theorem 3.9.
The collection of functions(ψn,k;(n,k)∈I) constitutes a Schauder basis of functions on C0(U,ℝd), that is, any element of C0(U,ℝd) can be written in a unique way as a sum of coefficients an,k multiplied by ψn,k.This theorem provides us with a complementary view of stochastic processes: in addition to the standard sample paths view, this structure allows to see the Gauss-Markov processes as coefficients on the computed basis. This duality is developed in the sequel.


## 3.2. The Sample Paths Space

### 3.2.1. The Construction Application

The Schauder basis of functions with compact supports constructed allows to define functions by considering the coefficients on this basis, which constitute sequences of real numbers in the space(3.47)ξΩ={ξ={ξn,k}I;∀(n,k)∈I,ξn,k∈Rd}=(Rd)I.
We equip ξΩ with the uniform norm ∥ξ∥∞=sup(n,k)∈ℐ|ξn,k|, where we write |ξn,k|=sup0≤i<d|(ξn,k)i|. We denote by ℬ(ξΩ) the Borelian sets of the topology induced by the uniform norm and we recall that C(ξΩ), the cylinder sets of ξΩ, form a generative family of Borelian sets. Remark that not any sequence of coefficients provides a continuous function, and one needs to assume a certain decrease in the coefficients to get convergence. A sufficient condition to obtain convergent sequences is to consider coefficients in the space(3.48)ξΩ′={ξ∈ξΩ∣∃δ∈(0,1),∃N∈N,∀(n,k)∈I∖IN,|ξn,k|<2nδ/2}.
This set is clearly a Borelian set of ξΩ  since it can be written as a countable intersection and union of cylinder, namely, by denoting by 𝒥 the set of finite subset of ℕ and δp=1-1/p, p>1,(3.49)ξΩ′=⋃p>1⋃J∈J⋂n∈N∖J{ξ∈ξΩ∣max⁡0≤k<2n-1|ξn,k|<2nδp/2}.
It is also easy to verify that it forms a vectorial subspace of ξΩ.After these definitions, we are in position to introduce the following useful function.Definition 3.10.
One denotes byΨN the partial construction application:
(3.50)ΨN={ξΩ,⟶C0([0,1],Rd),ξ,⟼∑(n,k)∈INψn,k(t)⋅ξn,k,
where the C0([0,1],ℝd) is the d-dimensional Wiener space, which is complete under the uniform norm ∥x∥∞=sup0≤t≤1|x(t)|.This sequence of partial construction applications is shown to converge to the construction application in the following.Proposition 3.11.
For everyξ in ξΩ′, ΨN(ξ) converges uniformly toward a continuous function in C0([0,1],ℝd). One will denote this function Ψ(ξ), defined as
(3.51)Ψ:{ξΩ′⟶C0([0,1],Rd),ξ⟼∑(n,k)∈Iψn,k(t)⋅ξn,k,
and this application will be referred to as the construction application.This proposition is proved in AppendixD. The image of this function constitutes a subset of the Wiener space continuous functions C0([0,1],ℝd). Let us now define the vectorial subspace xΩ′=Ψ(ξΩ′) of C0([0,1],ℝd) so that Ψ appears as a bijection.It is important to realize that, in the multidimensional case, the spacexΩ′ depends on Γ and α in a nontrivial way. For instance, assuming that α=0, the space xΩ′ depends obviously crucially on the rank of Γ. To fix the idea, for a given constant Γ(t)=[0,0⋯1]T in ℝd×1, we expect the space xΩ′ to only include sample paths of C0([0,1],ℝd) for which the n-1 first components are constant. Obviously, a process with such sample paths is degenerated in the sense that its covariance matrix is not invertible.Yet, if we additionally relax the hypothesis thatα≠0, the space xΩ′ can be dramatically altered: if we take(3.52)α(t)=[01⋱⋱⋱10],
the space xΩ′ will represent the sample space of the d-1-integrated Wiener process, a nondegenerate d-dimensional process we fully develop in the example section.However, the situation is much simpler in the one-dimensional case: because the uniform convergence of the sample paths is preserved as long asα is continuous and Γ is nonzero through (D.8),  the definition xΩ′ does not depend on α or Γ. Moreover, in this case, the space xΩ′ is large enough to contain reasonably regular functions as proved in Appendix D, Proposition 3.In the case of thed-1-integrated Wiener process, the space xΩ′ clearly contains the functions {f=(fd-1,…,f0)∣f0∈H,fi′=fi-1,0<i<d}.This remark does not hold  that the spacexΩ′ does not depend on α as long as α is continuous  because the uniform convergence of the sample paths is preserved through the change of basis of expansion ψn,k through (D.8).We equip the spacexΩ′ with the topology induced by the uniform norm on C0([0,1],ℝd). As usual, we denote ℬ(xΩ′) the corresponding Borelian sets. We prove in Appendix D the following.Proposition 3.12.
The functionΨ:(ξΩ′,ℬ(ξΩ′))→(xΩ′,ℬ(xΩ′)) is a bounded continuous bijection.We therefore conclude that we dispose of a continuous bijection mapping the coefficients onto the sample paths,Ψ. We now turn to study its inverse, the coefficient application, mapping sample paths on coefficients over the Schauder basis.


### 3.2.2. The Coefficient Application

In this section, we introduce and study the properties of the following function.Definition 3.13.
One callscoefficient application and denotes by Ξ the function defined by
(3.53)Ξ:{C0([0,1],Rd)⟶Ωξ=(Rd)I,x⟼Δ(x)={Δ(x)}(n,k)∈Iwith{Δ(x)}n,k=P(δn,k,x).Should a functionx admit a uniformly convergent decomposition in terms of the basis of elements ψn,k, the function Δ gives its coefficients in such a representation. More precisely, we have the following.Theorem 3.14.
The functionΔ:(xΩ′,ℬ(xΩ′))→(ξΩ′,ℬ(ξΩ′)) is a measurable linear bijection whose inverse is Ψ=Δ-1.The proof of this theorem is provided in AppendixD.


## 3.2.1. The Construction Application

The Schauder basis of functions with compact supports constructed allows to define functions by considering the coefficients on this basis, which constitute sequences of real numbers in the space(3.47)ξΩ={ξ={ξn,k}I;∀(n,k)∈I,ξn,k∈Rd}=(Rd)I.
We equip ξΩ with the uniform norm ∥ξ∥∞=sup(n,k)∈ℐ|ξn,k|, where we write |ξn,k|=sup0≤i<d|(ξn,k)i|. We denote by ℬ(ξΩ) the Borelian sets of the topology induced by the uniform norm and we recall that C(ξΩ), the cylinder sets of ξΩ, form a generative family of Borelian sets. Remark that not any sequence of coefficients provides a continuous function, and one needs to assume a certain decrease in the coefficients to get convergence. A sufficient condition to obtain convergent sequences is to consider coefficients in the space(3.48)ξΩ′={ξ∈ξΩ∣∃δ∈(0,1),∃N∈N,∀(n,k)∈I∖IN,|ξn,k|<2nδ/2}.
This set is clearly a Borelian set of ξΩ  since it can be written as a countable intersection and union of cylinder, namely, by denoting by 𝒥 the set of finite subset of ℕ and δp=1-1/p, p>1,(3.49)ξΩ′=⋃p>1⋃J∈J⋂n∈N∖J{ξ∈ξΩ∣max⁡0≤k<2n-1|ξn,k|<2nδp/2}.
It is also easy to verify that it forms a vectorial subspace of ξΩ.After these definitions, we are in position to introduce the following useful function.Definition 3.10.
One denotes byΨN the partial construction application:
(3.50)ΨN={ξΩ,⟶C0([0,1],Rd),ξ,⟼∑(n,k)∈INψn,k(t)⋅ξn,k,
where the C0([0,1],ℝd) is the d-dimensional Wiener space, which is complete under the uniform norm ∥x∥∞=sup0≤t≤1|x(t)|.This sequence of partial construction applications is shown to converge to the construction application in the following.Proposition 3.11.
For everyξ in ξΩ′, ΨN(ξ) converges uniformly toward a continuous function in C0([0,1],ℝd). One will denote this function Ψ(ξ), defined as
(3.51)Ψ:{ξΩ′⟶C0([0,1],Rd),ξ⟼∑(n,k)∈Iψn,k(t)⋅ξn,k,
and this application will be referred to as the construction application.This proposition is proved in AppendixD. The image of this function constitutes a subset of the Wiener space continuous functions C0([0,1],ℝd). Let us now define the vectorial subspace xΩ′=Ψ(ξΩ′) of C0([0,1],ℝd) so that Ψ appears as a bijection.It is important to realize that, in the multidimensional case, the spacexΩ′ depends on Γ and α in a nontrivial way. For instance, assuming that α=0, the space xΩ′ depends obviously crucially on the rank of Γ. To fix the idea, for a given constant Γ(t)=[0,0⋯1]T in ℝd×1, we expect the space xΩ′ to only include sample paths of C0([0,1],ℝd) for which the n-1 first components are constant. Obviously, a process with such sample paths is degenerated in the sense that its covariance matrix is not invertible.Yet, if we additionally relax the hypothesis thatα≠0, the space xΩ′ can be dramatically altered: if we take(3.52)α(t)=[01⋱⋱⋱10],
the space xΩ′ will represent the sample space of the d-1-integrated Wiener process, a nondegenerate d-dimensional process we fully develop in the example section.However, the situation is much simpler in the one-dimensional case: because the uniform convergence of the sample paths is preserved as long asα is continuous and Γ is nonzero through (D.8),  the definition xΩ′ does not depend on α or Γ. Moreover, in this case, the space xΩ′ is large enough to contain reasonably regular functions as proved in Appendix D, Proposition 3.In the case of thed-1-integrated Wiener process, the space xΩ′ clearly contains the functions {f=(fd-1,…,f0)∣f0∈H,fi′=fi-1,0<i<d}.This remark does not hold  that the spacexΩ′ does not depend on α as long as α is continuous  because the uniform convergence of the sample paths is preserved through the change of basis of expansion ψn,k through (D.8).We equip the spacexΩ′ with the topology induced by the uniform norm on C0([0,1],ℝd). As usual, we denote ℬ(xΩ′) the corresponding Borelian sets. We prove in Appendix D the following.Proposition 3.12.
The functionΨ:(ξΩ′,ℬ(ξΩ′))→(xΩ′,ℬ(xΩ′)) is a bounded continuous bijection.We therefore conclude that we dispose of a continuous bijection mapping the coefficients onto the sample paths,Ψ. We now turn to study its inverse, the coefficient application, mapping sample paths on coefficients over the Schauder basis.


## 3.2.2. The Coefficient Application

In this section, we introduce and study the properties of the following function.Definition 3.13.
One callscoefficient application and denotes by Ξ the function defined by
(3.53)Ξ:{C0([0,1],Rd)⟶Ωξ=(Rd)I,x⟼Δ(x)={Δ(x)}(n,k)∈Iwith{Δ(x)}n,k=P(δn,k,x).Should a functionx admit a uniformly convergent decomposition in terms of the basis of elements ψn,k, the function Δ gives its coefficients in such a representation. More precisely, we have the following.Theorem 3.14.
The functionΔ:(xΩ′,ℬ(xΩ′))→(ξΩ′,ℬ(ξΩ′)) is a measurable linear bijection whose inverse is Ψ=Δ-1.The proof of this theorem is provided in AppendixD.


## 4. Representation of Gauss-Markov Processes

### 4.1. Inductive Construction of Gauss-Markov Processes

Up to this point, we have rigorously defined  the dual spaces of sample pathsxΩ′ and coefficients ξΩ′. Through the use of the Schauder basis ψn,k and its dual family of generalized functions δn,k, we have defined the inverse measurable bijections Ψ and Δ transforming one space into the other. In doing so, we have unraveled the fundamental role played by the underlying orthonormal basis ϕn,k. We now turn to use this framework to formulate a pathwise construction of the Gauss-Markov processes in the exact same flavor as the Lévy-Ciesielski construction of the Wiener process.


#### 4.1.1. Finite-Dimensional Approximations

Considering the infinite-dimensional subspacexΩ′ of C0([0,1],ℝd), let us introduce the equivalence relation ~N as(4.1)x∼Ny⟺∀t∈DN,x(t)=y(t).
We can use the functions Ψ to carry through the structure of ~N on the infinite-dimensional space of coefficients ξΩ′:(4.2)ξ∼Nη⟺Ψ(ξ)∼NΨ(η)⟺∀(n,k)∈IN,ξn,k=ηn,k,
which clearly entails that x~Ny if and only if Δ(x)~NΔ(y). We denote the sets of equivalence classes of xΩ′/~N=ΩxN and ξΩ′/~N=ΩξN, which are both clearly isomorphic xΩN=(ℝd)ℐ=ΩξN. For every N>0, we define the finite-dimensional operators ΨN=ixN∘Ψ∘pξN and ΔN=iξN∘Δ∘pxN, with the help of the canonical projections ξpN:Ωξ′→ΩξN, xpN:Ωx′→ΩxN and the inclusion map iξN:ΩξN→Ωξ′, ixN:ΩxN→Ωx′.The results of the preceding sections straightforwardly extend on the equivalence classes, and in particular we see that the functionsΨN:ΩξN→ΩxN and ΔN:ΩxN→ΩξN are linear finite-dimensional bijections satisfying ΨN=ΔN-1. We write e={ep,q}(p,q)∈ℐ (resp., f={fp,q}(p,q)∈ℐ), the canonical basis of ξΩN (resp., ΩxN) when listed in the recursive dyadic order. In these bases, the matrices ΨN and ΔN are lower block triangular. Indeed, denoting ΨN in the natural bases e={ep,q}(p,q)∈ℐ and f={fp,q}(p,q)∈ℐ by(4.3)ΨN=[ψn,k(mi,j)]=[Ψn,ki,j],
where Ψn,ki,j is a d×d matrix, the structure of the nested support Sn,k entails the block-triangular structure (where only possibly nonzero coefficients are written):(4.4)ΨαN=[ψ0,00,0ψ0,01,0ψ1,01,0ψ0,02,0ψ1,02,0ψ2,02,0ψ0,02,1ψ1,02,1ψ2,12,1ψ0,03,0ψ1,03,0ψ2,03,0ψ3,03,0ψ0,03,1ψ1,03,1ψ2,03,1ψ3,13,1ψ0,03,2ψ1,03,2ψ2,13,2ψ3,23,2ψ0,03,3ψ1,03,3ψ2,13,3ψ3,33,3⋮⋱].Similarly, for the matrix representation ofΔN in the natural bases en,k and fi,j(4.5)ΔN=[Δi,jn,k]
proves to have the following triangular form:(4.6)ΔN=[g-1(t0,0)TM0,0-g-1(t0,0)TR1,0g-1(t1,0)TM1,0-g-1(t0,0)TR2,0g-1(t1,0)TM2,0-g-1(t0,0)TR2,1-g-1(t1,0)TL2,1g-1(t2,1)TM2,1⋮⋱].
The duality property, Proposition 3.8, simply reads for all 0≤n<N and 0≤k<2n-1, 0≤p<N and 0≤p<2q-1(4.7)P(δp,q,ψn,k)=∑(n,k)∈INΔi,jp,q⋅Ψn,ki,j=δn,kp,qId,
that is, ΔN·ΨN=IdξΩN.  But because  we are now in a finite-dimensional setting, we also have ΨN·ΔN=IdxΩN:(4.8)δk,li,jId=∑(p,q)∈INΨp,qi,j⋅Δk,lp,q.
Realizing that δk,li,jId represents the class of functions x in xΩ′ whose values are zero on every dyadic point  of DN except for x(l2k)=Id, {Δk,lp,q}(p,q)∈ℐN clearly appear as the coefficients of the decomposition of such functions in the bases ψp,q for (p,q) in ℐN.DenotingΞ={Ξn,k}(n,k)∈I, a set of independent Gaussian variables of law 𝒩(0,Id) on (Ω,ℱ,P), and for all N>0, we form the finite dimensional Gauss-Markov vector [Xi,jN](i,j)∈ℐN as(4.9)Xi,jN=∑(n,k)∈INψn,k(mi,j)⋅Ξn,k,
which, from Corollary 2.5, has the same law as [Xt]t∈DN, the finite-dimensional random vector obtained from sampling X on DN (modulo a permutation on the indices). We then prove the following lemma that sheds light on the meaning of the construction.Lemma 4.1.
The Cholesky decomposition of the finite-dimensional covariance block matrixΣN is given by ΣN=ΨN·ΨNT.Proof.
For every0≤t,s≤1, we compute the covariance of the finite-dimensional process XN as
(4.10)CN(t,s)=E[XtN⋅(XsN)T]=∑n=0N∑0≤k<2n-1ψn,k(t)⋅(ψn,k(s))T.
From there, we write the finite-dimensional covariance block matrix ΣN in the recursively ordered basis fi,j for 0≤i≤N, 0≤j<2i-1, as
(4.11)[𝚺N]k,li,j=CN(mi,j,mk,l)=∑n=0N∑0≤k<2n-1Ψn,ki,j⋅ψn,kk,l.
We already established that the matrix ΨN was triangular with positive diagonal coefficient, which entails that the preceding equality provides us with the Cholesky decomposition of Σ.In the finite-dimensional case, the inverse covariance or potential matrix is a well-defined quantity and we straightforwardly have the following corollary.Corollary 4.2.
The Cholesky decomposition of the finite-dimensional inverse covariance matrixΣN-1 is given by ΣN-1=ΔNT·ΔN.Proof.
The result stems for the equalitiesΣN-1=(ΨN·ΨNT)-1=(ΨN-1)T·ΨN-1=ΔNT·ΔN.


#### 4.1.2. The Lévy-Ciesielski Expansion

We now show that, asymptotically, the basesψn,k allow  us to faithfully build the Gauss-Markov process from which we have derived its expression. In this perspective we consider Ξ={Ξn,k}(n,k)∈ℐ, a set of independent Gaussian variables of law 𝒩(0,Id) on (Ω,ℱ,P), and, for all N>0, we form the finite-dimensional continuous Gaussian process ZN, defined for 0≤t≤1 by(4.12)XtN=∑(n,k)∈INψn,k(t)⋅Ξn,k,
which, from the result of Theorem 2.4, has the same law ZtN=𝔼[Xt∣ℱN]. We prove the following lemma.Lemma 4.3.
The sequence of processesXN almost surely converges towards a continuous Gaussian process denoted by X∞.Proof.
For all fixedN>0 and for any ω in Ω, we know that t↦XtN(ω) is continuous. Moreover, we have established, that, for every ξ in ξΩ′, XN(ξ) converges uniformly in t toward a continuous limit denoted by XN(ξ). Therefore, in order to prove that limN→∞XN defines almost surely a process X with continuous paths, it is sufficient to show that Pξ(ξΩ′)=1, where Pξ=PΞ-1 is the Ξ-induced measure on ξΩ, which stems from a classical Borel-Cantelli argument. For ξ a random variable of normal law 𝒩(0,1), and a>0, we have
(4.13)P(|ξ|>a)=2π∫a∞e-u2/2du≤2π∫a∞uae-u2/2du=2πe-a2/2a.
Then, for any δ>0(4.14)Pξ(max⁡0≤k<2n-1|ξn,k|∞>2nδ/2)≤d2nP(|ξ|>d2nδ/2)=2π2(1-δ/2)nexp⁡(-2nδ-1).
Since the series
(4.15)∑n=0∞2π2(1-δ/2)nexp⁡(-2nδ-1)
is convergent, the Borel-Cantelli argument implies that Pξ(ξΩ′)=1. Eventually, the continuous almost-sure limit process Xt∞ is Gaussian as a countable sum of Gaussian processes.Now that these preliminary remarks have been made, we can evaluate, for anyt and s in [0,1], the covariance of X as the limit of the covariance of XN.Lemma 4.4.
For any0≤t,s≤1, the covariance of X∞={Xt∞=Ψt∘Ξ;0≤t≤1} is
(4.16)C(t,s)=E[Xt∞⋅(Xs∞)T]=g(t)h(t∧s)g(s)T.Proof.
AsΞn,k are independent Gaussian random variables of normal law 𝒩(0,Id), we see that the covariance of XN is given by
(4.17)CN(t,s)=E[XtN⋅(XsN)T]=∑(n,k)∈INψn,k(t)⋅(ψn,k(s))T.
To compute the limit of the right-hand side, we need to remember that the element of the bases ψn,k and the functions ϕn,k are linked by the following relation:
(4.18)ψn,k(t)=K[ϕn,k]=g(t)∫U1[0,t](s)f(s)ϕn,k(s)ds,
from which we deduce
(4.19)CN(t,s)=g(t)(∑(n,k)∈IN(∫U1[0,t](u)f(u)ϕn,k(u)du)(∫U1[0,s](v)f(v)ϕn,k(v)dv)T)g(s)T.
Defining the auxiliary ℝd×d-valued function
(4.20)κn,k(t)=∫U1[0,t](u)f(u)ϕn,k(u)du,
we observe that the (i,j)-coefficient function reads
(4.21)(κn,k)i,j(t)=∫U1[0,t](u)(li(f(u))T⋅cj(ϕn,k(u)))du=∫U(1[0,t](u)ci(fT(u))T⋅cj(ϕn,k(u)))du,
where 1[0,t] is the real function that is one if 0≤u≤t and zero otherwise. As we can write
(4.22)1[0,t](u)ci(fT(u))=fT(u)⋅[0⋮1[0,t](u)⋮0]⟵i,
we see that the function fi,t=1[0,t]ci(fT) belongs to Lf2, so that we can write (κn,k)i,j(t) as a scalar product in the Hilbert space Lf2:
(4.23)(κn,k)i,j(t)=∫Ufi,tT(u)⋅cj(ϕn,k(u))du=(fi,t,cj(ϕn,k)).
We then specify the (i,j)-coefficient of g-1(t)CN(t,s)(g-1(s))T writing
(4.24)∑(n,k)∈IN(κ(t)⋅κ(s)T)i,j=∑(n,k)∈IN∑p=0d-1(fi,t,cj(ϕn,k))(fj,s,cj(ϕn,k)),
and, remembering that the family of functions cj(ϕn,k) forms a complete orthonormal system of Lf2, we can use the Parseval identity, which reads
(4.25)∑(n,k)∈I(κ(t)⋅κ(s)T)i,j=(fi,t,fj,s)=∫U1[0,t](u)ci(fT(u))T⋅1[0,s](u)cj(fT(u))du=∫0t∧s(f⋅fT)i,j(u)du.
Thanks to this relation, we can conclude the evaluation of the covariance since
(4.26)lim⁡N→∞CN(t,s)=g(t)(∫0t∧s(f⋅fT)(u)du)g(s)T=g(t)h(t∧s)g(s)T.We stress the fact that the relation(4.27)C(t,s)=∑(n,k)∈Iψn,k(t)⋅(ψn,k(s))T=Ψ(t)∘ΨT(s)
provides us with a continuous version of the Cholesky decomposition of the covariance kernel C. Indeed, if we chose σn,k as the Cholesky square root of Σn,k, we remark that the operators Φ are triangular in the following sense: consider the chain of nested vectorial spaces {Fn,k}(n,k)∈ℐ(4.28)F0,0⊂F1,0⊂(F2,0⊂F2,1)⋯⊂(Fn,0⊂⋯⊂Fn,2n-1)⋯⊂Ωξ′
with Fn,k=span{fi,j∣0≤i≤n,0≤j≤k}; then, for every (n,k) in ℐ, the operator Ψ transforms the chain {Fn,k}(n,k)∈ℐ into the chain(4.29)E0,0⊂E1,0⊂(E2,0⊂E2,1)⋯⊂(En,0⊂⋯⊂En,2n-1)⋯⊂xΩ′
with ℰn,k=span{Ψi,j∣0≤i≤n,0≤j≤k}.The fact that this covariance is equal to the covariance of the processX, solution of (2.1) implies that we have the following fundamental result.Theorem 4.5.
The processX∞ is equal in law to the initial Gauss-Markov process X used to construct the basis of functions.Remark 4.6.
Our multiresolution representation of the Gauss-Markov processes appears to be the direct consequence of the fact that, because of the Markov property, the Cholesky decomposition of the finite-dimensional covariance admits a simple inductive continuous limit. More generally, triangularization of the kernel operators has been studied in depth [28, 40–42], and it would be interesting to investigate if these results make possible a similar multiresolution approach for non-Markov Gaussian processes. In this regard, we naturally expect to lose the compactness of the supports of a putative basis.Remark 4.7.
We eventually underline the fact that large deviations related to this convergence can be derived through the use of the Baldi and Caramellino good rate function related to the Gaussian pinned processes [43, 44].


### 4.2. Optimality Criterion of the Decomposition

In the following, we draw from the theory of interpolating splines to further characterize the nature of our proposed basis for the construction of the Gauss-Markov processes. Essentially adapting the results from the previous works [45, 46], we first show that the finite-dimensional sample paths of our construction induce a nested sequence ℰN of the reproducing Hilbert kernel space (RKHS). In turn, the finite-dimensional process XN naturally appears as the orthogonal projection of the infinite-dimensional process X onto ℰN. We then show that such an RKHS structure allows us to define a unicity criterion for the finite-dimensional sample path as the only functions of ℰ that minimize a functional, called Dirichlet energy, under constraint of interpolation on DN (equivalent to conditioning on the times DN). In this respect, we point out that the close relation between the Markov processes and the Dirichlet forms is the subject of a vast literature, largely beyond the scope of the present paper (see, e.g., [18]).


#### 4.2.1. Sample Paths Space as a Reproducing Hilbert Kernel Space

In order to define the finite-dimensional sample paths as a nested sequence of RKHSs, let us first define the infinite-dimensional operator(4.30)Φ:{l2(ξΩ)⟼Lf2,ξ⟼Φ[ξ]={t⟼∑(n,k)∈Iϕn,k(t)⋅ξn,k}.
Since we know that the column functions of ϕn,k form a complete orthonormal system of Lf2, the operator Φ is an isometry and its inverse satisfies Φ-1=ΦT, which reads for all v in Lf2(4.31)[Φ-1[v]]n,k=∫Uϕn,kT(t)⋅v(t)dt=P(ϕn,k,v).
Equipped with this infinite-dimensional isometry, we then consider the linear operator ℒ=Φ∘Δ suitably defined on the set(4.32)E={u∈C0(U,Rd)∣L[u]∈Lf2}={u∈C0(U,Rd)∣Δ[u]∈l2(ξΩ)}
with ∥ξ∥22=∑n,k∈ℐ|ξn,k|22, the l2 norm of ξΩ. The set ℰ form an infinite-dimensional vectorial space that is naturally equipped with the inner product(4.33)∀(u,v)∈E2,〈u,v〉=∫UL[u](t)T⋅L[v](t)dt=(L[u],L[v]).
Moreover since u(0)=v(0)=0, such an inner product is definite positive and,  consequently, ℰ forms an Hilbert space.Remark 4.8.
Two straightforward remarks are worth making. First, the spaceℰ is strictly included in the infinite-dimensional sample paths space xΩ′. Second, notice that, in the favorable case m=d, if f is everywhere invertible with continuously differentiable inverse, we have ℒ=𝒟=𝒦-1. More relevantly, the operator ℒ can actually consider a first-order differential operator from ℰ to Lf2 as a general left inverse of the integral operator 𝒦. Indeed, realizing that on Lf2,𝒦 can be expressed as 𝒦=Ψ∘Φ-1, we clearly have
(4.34)L∘K=Φ∘Δ∘Ψ∘Φ-1=IdLf2.We know motivate the introduction of the Hilbert spaceℰ by the following claim.Proposition 4.9.
The Hilbert space(ℰ,〈,〉) is a reproducing kernel Hilbert space (RKHS) with ℝd×d-valued reproducing kernel C, the covariance function of the process X.Proof.
Consider the problem of finding all elementsu of ℰ solution of the equation ℒ[u]=v for v in Lf2. The operator 𝒦 provides us with a continuous ℝd×m-valued kernel function k:
(4.35)∀(t,s)∈U2,k(t,s)=1[0,t](s)g(t)⋅f(s),
which is clearly the Green function for our differential equation. This entails that the following equalitiy holds for every u in ℰ:
(4.36)u(t)=∫Uk(t,s)L[u](s)ds.
Moreover, we can decompose the kernel k in the Lf2 sense as
(4.37)k(t,s)=∑(n,k)∈Iψn,k(t)⋅ϕn,kT(s)
since we have
(4.38)k(t,s)=K[δsIdLf2](t)=K[∑(n,k)∈Iϕn,k⋅ϕn,kT(s)](t)=∑(n,k)∈IK[ϕn,k](t)⋅ϕn,kT(s),
with  δs=δ(·-s). Then, we clearly have
(4.39)C(t,s)=∫Uk(t,u)⋅k(s,u)Tdu=∑(n,k)∈Iψn,k(t)⋅ψn,kT(s),
where we recognize the covariance function of X, which implies
(4.40)k(t,s)=∑(n,k)∈Iψn,k(t)⋅L[ϕn,k]T(s)=L[C(t,⋅)].
Eventually, for all u in Lf2, we have
(4.41)u(t)=∫UL[C(t,⋅)](s)⋅L[u](s)ds=P〈C(t,⋅),u〉,
where we have introduced the 𝒫-operator associated with the inner product 〈,〉: for all ℝd×d-valued functions A and B defined on U such that the columns ci(A) and ci(B), 0≤i<d, are in ℰ, we define the matrix 𝒫〈A,B〉 in ℝd×d by
(4.42)∀0≤i,j<d,P〈A,B〉i,j=〈ci(A),cj(B)〉.
By the Moore-Aronszajn theorem [47], we deduce that there is a unique reproducing kernel Hilbert space associated with a given covariance kernel. Thus, ℰ is the reproducing subspace of C0(U,ℝd) corresponding to the kernel C, with respect to the inner product 〈,〉.Remark 4.10.
From a more abstract point of view, it is well known that the covariance operator of a Gaussian measure defines an associated Hilbert structure [48, 49].In the sequel, we will use the spaceℰ as the ambient Hilbert space to define the finite-dimensional sample paths spaces as a nested sequence of RKHS. More precisely, let us write for ℰN the finite-dimensional subspace of ℰ(4.43)EN={u∈C0(U,Rd)∣L[u]∈Lf,N2},
with the space Lf,N2 being defined as(4.44)Lf,N2=span⁡[{ci(ϕn,k)}n,k∈IN,0≤i<d].We refer to such spaces as finite-dimensional approximation spaces  since we remark that(4.45)EN=span⁡[{ci(ψn,k)}n,k∈IN,0≤i<d]=ΨN[ξΩN],
which means that the space ℰN is made of the sample space of the finite-dimensional process XN. The previous definition makes obvious the nested structure ℰ0⊂ℰ1⊂⋯⊂ℰ, and it is easy to characterize each space ℰN as a reproducing Hilbert kernel space.Proposition 4.11.
The Hilbert spaces(ℰN,〈,〉) are reproducing kernel Hilbert spaces (RKHSs) with ℝd×d-valued reproducing kernel CN, the covariance function of the process XN.Proof.
The proof this proposition follows the exact same argument as that in the case ofℰ, but with the introduction of finite-dimensional kernels kN(4.46)∀(t,s)∈U2,kN(t,s)=∑(n,k)∈INψn,k(t)⋅ϕn,kT(s),
and the corresponding covariance function
(4.47)∀(t,s)∈[0,1]2,CN(t,s)=∑(n,k)∈INψn,k(t)⋅ψn,kT(s).


#### 4.2.2. Finite-Dimensional Processes as Orthogonal Projections

The framework set in the previous section offers a new interpretation of our construction. Indeed, for allN>0, the columns of {ψn,k}(n,k)∈ℐN form an orthonormal basis of the space ℰN:(4.48)P〈ψn,k,ψp,q〉=P(L[ψn,k],L[ψp,q])=P(ϕn,k,ϕp,q)=δp,qn,k.
This leads to defining the finite-dimensional approximation xN of an sample path x of ℰ as the orthogonal projection of x on ℰN with respect to the inner product 〈,〉. At this point, it is worth remembering that the space ℰ is strictly contained in xΩ′ and does not coincide with xΩ′: actually one can easily show that P(ℰ)=0. We devote the rest of this section to defining the finite-dimensional processes ZN=𝔼N[X] resulting from the conditioning on DN, as pathwise orthogonal projection of the original process X on the sample space ℰN.Proposition 4.12.
For anyN>0, the conditioned processes 𝔼N[X] can be written as the orthogonal projection of X on ℰN with respect to 〈,〉:
(4.49)EN[X]=∑(n,k)∈INψn,k⋅P〈ψn,k,X〉.The only hurdle to prove Proposition4.12 is purely technical in the sense that the process X exists in a larger space than ℰ: we need to find a way to extend the definition of 〈,〉 so that the expression bears a meaning. Before answering this point quite straightforwardly, we need to establish the following lemma.Lemma 4.13.
Writing the Gauss-Markov processXt=∫01k(t,s)dWs, for all N>0, the conditioned process ZN=𝔼N[X] is expressed as the stochastic integral
(4.50)ZN=∫01kN(t,s)dWswithkN(t,s)=∑(n,k)∈INψn,k(t)⋅ϕn,kT(s).Proof.
In the previous section, we have noticed that the kernelkN converges toward the kernel k (the Green function) in the Lf2 sense
(4.51)k(t,s)=∑(n,k)∈Iψn,k(t)⋅ϕn,kT(s)=lim⁡N→∞∑(n,k)∈INψn,k(t)⋅ϕn,kT(s)kN=lim⁡N→∞kN(t,s).
This implies that the process X as the stochastic integral  can also be written as
(4.52)Xt=g(t)∫U1[0,t](s)(s)dWs=∫01k(t,s)dWs=lim⁡N→∞∫01kN(t,s)dWs.
Specifying the decomposition of kN, we can then naturally express X as the convergent sum
(4.53)Xt=∑(n,k)∈Iψn,k⋅Ξn,kwithΞn,k=∫01ϕn,kT(s)dWs,
where the orthonormality property of the ϕn,k with respect to (,) makes the vectors Ξn,k appear as independent d-dimensional Gaussian variables of law 𝒩(0,Id). It is then easy to see that by definition of the elements ψn,k, for almost every ω in Ω, we then have
(4.54)∀N>0,0≤t≤1,ZN(ω)=EN[X](ω)=∑(n,k)∈INψn,k⋅Ξn,k(ω),
and we finally recognize in the previous expression that for all 0≤t≤1(4.55)ZtN=∑(n,k)∈INψn,k⋅Ξn,k=∑(n,k)∈INψn,k(t)⋅∫Uϕn,kT(s)dWs=∫01kN(t,s)dWs.We can now proceed to justify the main result of Proposition4.12.Proof.
The finite-dimensional processesZN defined through Lemma 4.13 have sample paths t↦ZtN(ω) belonging to ℰN. Moreover, for almost every ω in Ω  and for all n,k in ℐN,
(4.56)P〈ψn,k,ZN(ω)〉=P〈ψn,k,∫01kN(t,s)dWs(ω)〉=P〈ψn,k,∑(p,q)∈INψp,q(ω)⋅∫01ϕp,qT(s)dWs(ω)〉=∫01ϕn,kT(s)dWs(ω),
because of the orthonormality property of ψn,k with respect to 〈,〉. As the previous equalities hold  for every N>0, the applications x↦𝒫〈ψn,k,x〉 can naturally be extended on xΩ′ by continuity. Therefore, it makes sense to write, for all (n,k) in ℐN, 𝒫〈ψn,k,ZN〉=limN→∞𝒫〈ψn,k,ZN〉=def𝒫〈ψn,k,X〉 even if   X is defined into a larger sample space than ℰ. In other words, we have
(4.57)P〈ψn,k,X〉=∫01ϕn,kT(s)dWs=Ξn,k,
and we can thus express the conditioned process ZN=𝔼N[X] as the orthogonal projection of X onto the finite sample path ℰN by writing
(4.58)ZN=∑(n,k)∈INψn,k⋅P〈ψn,k,X〉.


#### 4.2.3. Optimality Criterion of the Sample Paths

Proposition4.12 elucidates the structure of the conditioned processes ZN as pathwise orthogonal projections of X on the finite-dimensional RKHS ℰN. It allows us to cast the finite sample paths in a geometric setting and incidentally, to give a characterization of them as the minimizer of some functionals. In doing so, we shed a new light on well-known results of the interpolation theory [50–52] and extend them to the multidimensional case.The central point of this section reads as follows.Proposition 4.14.
Given a functionx in ℰ, the function xN=(Ψ∘ΔN)[x] belongs to ℰN and is defined by the following optimal criterion: xN is the only function in ℰ interpolating x on DN such that the functional
(4.59)〈y,y〉=‖L[y](t)‖22=∫01|L[y](t)|22dt
takes its unique minimal value over ℰ in xN.Proof.
The spaceℰN has been defined as ℰN=ΨN[ξΩN]=Ψ∘ΔN[ℰ] so that, for all x in ℰ, xN clearly belongs to ℰN. Moreover, xN interpolates x on DN: indeed, we know that the finite-dimensional operators ΔN and ΨN-1 are inverse of each other ΔN=ΨN-1, which entails that for all t in DN(4.60)xN(t)=(Ψ∘ΔN)[x](t)=(ΨN∘ΔN)[x](t)=x(t),
where we use the fact that, for any ξ in ξΩ′  and for all t in DN, ΨN[ξ](t)=Ψ[ξ](t) (recall that ψn,k(t)=0 if n>N and t belongs to DN).
Let us now show thatxN is determined in ℰ by the announced optimal criterion. Suppose y belongs to ℰ and interpolates x on DN, and remark that we can write
(4.61)〈y,y〉=‖L[y]‖22=‖(Φ∘Δ)[y](t)‖22=‖Δ[y]‖22,
since Φ is an isometry. Then, consider Δ[y] in l2(ξΩ) and remark that, since for all (n,k) in ℐN, δn,k are Dirac measures supported by DN, we have
(4.62)∀(n,k)∈IN,Δn,k[y]=P(δn,k,y)=P(δn,k,x)=Δn,k[x]=Δn,k[xN].
This entails
(4.63)‖Δ[y]‖22dt=∑(n,k)∈I|Δn,k[y]|22≥∑(n,k)∈IN|Δn,k[y]|22=‖Δ[xN]‖22dt.
Since, by definition of xN, δn,k[xN]=0 if n>N. Moreover, the minimum 〈xN,xN〉 is only attained for y such that δn,k[y]=0 if n>N and δn,k[y]=δn,k[x] if n≤N, which defines univocally xN. This establishes that, for all y in ℰ such that for all t in DN, y(t)=x(t), we have 〈xN,xN〉≤〈y,y〉 and the equality case holds if and only if y=xN.Remark 4.15.
Whenℒ represents a regular differential operator of order d, ∑i=1dai(t)Di, where D=d/dt, that is, for
(4.64)dXt=α(t)⋅Xt+Γ(t)⋅dWt,
with
(4.65)α(t)=[01⋱⋱⋱1adad-1…a1],Γ(t)=[00⋮1].
The finite-dimensional sample paths coincide exactly with the spline interpolation of order 2d+1, which are well known to satisfy the previous criterion [46]. This example will be further explored in the example section.The Dirichlet energy simply appears as the squared norm induced onℰ by the inner product 〈,〉, which in turn can be characterized as a Dirichlet quadratic form on ℰ. Actually, such a Dirichlet form can be used to define the Gauss-Markov process, extending the Gauss-Markov property to processes indexed on the multidimensional spaces parameter [19]. In particular, for an n-dimensional parameter space, we can condition such Gauss-Markov processes on a smooth n-1-dimensional boundary. Within the boundary, the sample paths of the resulting conditioned process (the solution to the prediction problem in [19]) are the solutions to the corresponding Dirichlet problems for the elliptic operator associated with the Dirichlet form.The characterization of the basis as the minimizer of such a Dirichlet energy (4.59) gives rise to an alternative method to compute the basis as the solution of a Dirichlet boundary value problem for an elliptic differential operator.Proposition 4.16.
Let us assume thatα and Γ are continuously differentiable and that Γ is invertible. Then, the functions μn,k are defined as
(4.66)μn,k(t)={μl(t),t∈[ln,k,mn,k],μr(t),t∈[mn,k,rn,k],0,else,
where μl and μr are the unique solutions of the second-order d-dimensional linear differential equation
(4.67)u′′+(Γ-1(αTΓ-Γ′)-α)u′-(Γ-1(αTΓ-Γ′)α+α′)u=0
with the following boundary value conditions:
(4.68)μl(ln,k)=0,μl(mn,k)=Id,μr(mn,k)=Id,μr(rn,k)=0.Proof.
By Proposition4.14, we know that μn,k(t) minimizes the convex functional
(4.69)∫01|L[u](s)|22ds
over ℰ, being equal to zero outside the interval [ln,k,rn,k] and equal to one at the point t=mn,k. Because of the hypotheses on α and Γ, we have ℒ=𝒟 and we can additionally restrain our search to functions that are twice continuously differentiable. Incidentally, we only need to minimize separately the contributions on the interval [ln,k,mn,k] and [mn,k,rn,k]. On both intervals, this problem is a classical Euler-Lagrange problem (see, e.g., [53]) and is solved using basic principles of calculus of variations. We easily identify the Lagrangian of our problem as
(4.70)L(t,u,u′)=|(u′-α(t)u(t))(Γ(t))-1|22=(u′(t)-α(t)u(t))T(Γ(t))-1(u′(t)-α(t)u(t)).
From there, after some simple matrix calculations, the Euler-Lagrange equations
(4.71)∂L(t,u,u′)∂ui-ddt(∂L(t,u,u′)∂u′i)=0,i=1,…,d,
can be expressed under the form:
(4.72)u′′+(Γ-1(αTΓ-Γ′)-α)u′-(Γ-1(αTΓ-Γ′)α+α′)u=0,
which ends the proof.Remark 4.17.
It is a simple matter of calculus to check that the expression ofμ given in Proposition 2.1 satisfies (4.67). Notice also that, in the case Γ=Id, the differential equation becomes
(4.73)u′′+(αT-α)u′-(αTα+α′)u=0,
which is further simplified for constant α.Under the hypotheses of Proposition4.16, we can thus define μn,k as the unique solution to the second-order linear differential equation (4.67) with the appropriate boundary values conditions. From this definition, it is then easy to derive the bases ψn,k by completing the following program.(1)
Compute thet↦μn,k(t) by solving the linear ordinary differential problem.(2)
Apply the differential operator𝒟 to get the functions 𝒟[μn,k].(3)
Orthonormalize the column functionst↦cj(𝒟[μn,k(t)]) by the Gram-Schmidt process.(4)
Apply the integral operator𝒦 to get the desired functions ψn,k (or equivalently multiply the original function t↦μn,k(t) by the corresponding Gram-Schmidt triangular matrix).Notice finally that each of these points is easily implemented numerically.


## 4.1. Inductive Construction of Gauss-Markov Processes

Up to this point, we have rigorously defined  the dual spaces of sample pathsxΩ′ and coefficients ξΩ′. Through the use of the Schauder basis ψn,k and its dual family of generalized functions δn,k, we have defined the inverse measurable bijections Ψ and Δ transforming one space into the other. In doing so, we have unraveled the fundamental role played by the underlying orthonormal basis ϕn,k. We now turn to use this framework to formulate a pathwise construction of the Gauss-Markov processes in the exact same flavor as the Lévy-Ciesielski construction of the Wiener process.


### 4.1.1. Finite-Dimensional Approximations

Considering the infinite-dimensional subspacexΩ′ of C0([0,1],ℝd), let us introduce the equivalence relation ~N as(4.1)x∼Ny⟺∀t∈DN,x(t)=y(t).
We can use the functions Ψ to carry through the structure of ~N on the infinite-dimensional space of coefficients ξΩ′:(4.2)ξ∼Nη⟺Ψ(ξ)∼NΨ(η)⟺∀(n,k)∈IN,ξn,k=ηn,k,
which clearly entails that x~Ny if and only if Δ(x)~NΔ(y). We denote the sets of equivalence classes of xΩ′/~N=ΩxN and ξΩ′/~N=ΩξN, which are both clearly isomorphic xΩN=(ℝd)ℐ=ΩξN. For every N>0, we define the finite-dimensional operators ΨN=ixN∘Ψ∘pξN and ΔN=iξN∘Δ∘pxN, with the help of the canonical projections ξpN:Ωξ′→ΩξN, xpN:Ωx′→ΩxN and the inclusion map iξN:ΩξN→Ωξ′, ixN:ΩxN→Ωx′.The results of the preceding sections straightforwardly extend on the equivalence classes, and in particular we see that the functionsΨN:ΩξN→ΩxN and ΔN:ΩxN→ΩξN are linear finite-dimensional bijections satisfying ΨN=ΔN-1. We write e={ep,q}(p,q)∈ℐ (resp., f={fp,q}(p,q)∈ℐ), the canonical basis of ξΩN (resp., ΩxN) when listed in the recursive dyadic order. In these bases, the matrices ΨN and ΔN are lower block triangular. Indeed, denoting ΨN in the natural bases e={ep,q}(p,q)∈ℐ and f={fp,q}(p,q)∈ℐ by(4.3)ΨN=[ψn,k(mi,j)]=[Ψn,ki,j],
where Ψn,ki,j is a d×d matrix, the structure of the nested support Sn,k entails the block-triangular structure (where only possibly nonzero coefficients are written):(4.4)ΨαN=[ψ0,00,0ψ0,01,0ψ1,01,0ψ0,02,0ψ1,02,0ψ2,02,0ψ0,02,1ψ1,02,1ψ2,12,1ψ0,03,0ψ1,03,0ψ2,03,0ψ3,03,0ψ0,03,1ψ1,03,1ψ2,03,1ψ3,13,1ψ0,03,2ψ1,03,2ψ2,13,2ψ3,23,2ψ0,03,3ψ1,03,3ψ2,13,3ψ3,33,3⋮⋱].Similarly, for the matrix representation ofΔN in the natural bases en,k and fi,j(4.5)ΔN=[Δi,jn,k]
proves to have the following triangular form:(4.6)ΔN=[g-1(t0,0)TM0,0-g-1(t0,0)TR1,0g-1(t1,0)TM1,0-g-1(t0,0)TR2,0g-1(t1,0)TM2,0-g-1(t0,0)TR2,1-g-1(t1,0)TL2,1g-1(t2,1)TM2,1⋮⋱].
The duality property, Proposition 3.8, simply reads for all 0≤n<N and 0≤k<2n-1, 0≤p<N and 0≤p<2q-1(4.7)P(δp,q,ψn,k)=∑(n,k)∈INΔi,jp,q⋅Ψn,ki,j=δn,kp,qId,
that is, ΔN·ΨN=IdξΩN.  But because  we are now in a finite-dimensional setting, we also have ΨN·ΔN=IdxΩN:(4.8)δk,li,jId=∑(p,q)∈INΨp,qi,j⋅Δk,lp,q.
Realizing that δk,li,jId represents the class of functions x in xΩ′ whose values are zero on every dyadic point  of DN except for x(l2k)=Id, {Δk,lp,q}(p,q)∈ℐN clearly appear as the coefficients of the decomposition of such functions in the bases ψp,q for (p,q) in ℐN.DenotingΞ={Ξn,k}(n,k)∈I, a set of independent Gaussian variables of law 𝒩(0,Id) on (Ω,ℱ,P), and for all N>0, we form the finite dimensional Gauss-Markov vector [Xi,jN](i,j)∈ℐN as(4.9)Xi,jN=∑(n,k)∈INψn,k(mi,j)⋅Ξn,k,
which, from Corollary 2.5, has the same law as [Xt]t∈DN, the finite-dimensional random vector obtained from sampling X on DN (modulo a permutation on the indices). We then prove the following lemma that sheds light on the meaning of the construction.Lemma 4.1.
The Cholesky decomposition of the finite-dimensional covariance block matrixΣN is given by ΣN=ΨN·ΨNT.Proof.
For every0≤t,s≤1, we compute the covariance of the finite-dimensional process XN as
(4.10)CN(t,s)=E[XtN⋅(XsN)T]=∑n=0N∑0≤k<2n-1ψn,k(t)⋅(ψn,k(s))T.
From there, we write the finite-dimensional covariance block matrix ΣN in the recursively ordered basis fi,j for 0≤i≤N, 0≤j<2i-1, as
(4.11)[𝚺N]k,li,j=CN(mi,j,mk,l)=∑n=0N∑0≤k<2n-1Ψn,ki,j⋅ψn,kk,l.
We already established that the matrix ΨN was triangular with positive diagonal coefficient, which entails that the preceding equality provides us with the Cholesky decomposition of Σ.In the finite-dimensional case, the inverse covariance or potential matrix is a well-defined quantity and we straightforwardly have the following corollary.Corollary 4.2.
The Cholesky decomposition of the finite-dimensional inverse covariance matrixΣN-1 is given by ΣN-1=ΔNT·ΔN.Proof.
The result stems for the equalitiesΣN-1=(ΨN·ΨNT)-1=(ΨN-1)T·ΨN-1=ΔNT·ΔN.


### 4.1.2. The Lévy-Ciesielski Expansion

We now show that, asymptotically, the basesψn,k allow  us to faithfully build the Gauss-Markov process from which we have derived its expression. In this perspective we consider Ξ={Ξn,k}(n,k)∈ℐ, a set of independent Gaussian variables of law 𝒩(0,Id) on (Ω,ℱ,P), and, for all N>0, we form the finite-dimensional continuous Gaussian process ZN, defined for 0≤t≤1 by(4.12)XtN=∑(n,k)∈INψn,k(t)⋅Ξn,k,
which, from the result of Theorem 2.4, has the same law ZtN=𝔼[Xt∣ℱN]. We prove the following lemma.Lemma 4.3.
The sequence of processesXN almost surely converges towards a continuous Gaussian process denoted by X∞.Proof.
For all fixedN>0 and for any ω in Ω, we know that t↦XtN(ω) is continuous. Moreover, we have established, that, for every ξ in ξΩ′, XN(ξ) converges uniformly in t toward a continuous limit denoted by XN(ξ). Therefore, in order to prove that limN→∞XN defines almost surely a process X with continuous paths, it is sufficient to show that Pξ(ξΩ′)=1, where Pξ=PΞ-1 is the Ξ-induced measure on ξΩ, which stems from a classical Borel-Cantelli argument. For ξ a random variable of normal law 𝒩(0,1), and a>0, we have
(4.13)P(|ξ|>a)=2π∫a∞e-u2/2du≤2π∫a∞uae-u2/2du=2πe-a2/2a.
Then, for any δ>0(4.14)Pξ(max⁡0≤k<2n-1|ξn,k|∞>2nδ/2)≤d2nP(|ξ|>d2nδ/2)=2π2(1-δ/2)nexp⁡(-2nδ-1).
Since the series
(4.15)∑n=0∞2π2(1-δ/2)nexp⁡(-2nδ-1)
is convergent, the Borel-Cantelli argument implies that Pξ(ξΩ′)=1. Eventually, the continuous almost-sure limit process Xt∞ is Gaussian as a countable sum of Gaussian processes.Now that these preliminary remarks have been made, we can evaluate, for anyt and s in [0,1], the covariance of X as the limit of the covariance of XN.Lemma 4.4.
For any0≤t,s≤1, the covariance of X∞={Xt∞=Ψt∘Ξ;0≤t≤1} is
(4.16)C(t,s)=E[Xt∞⋅(Xs∞)T]=g(t)h(t∧s)g(s)T.Proof.
AsΞn,k are independent Gaussian random variables of normal law 𝒩(0,Id), we see that the covariance of XN is given by
(4.17)CN(t,s)=E[XtN⋅(XsN)T]=∑(n,k)∈INψn,k(t)⋅(ψn,k(s))T.
To compute the limit of the right-hand side, we need to remember that the element of the bases ψn,k and the functions ϕn,k are linked by the following relation:
(4.18)ψn,k(t)=K[ϕn,k]=g(t)∫U1[0,t](s)f(s)ϕn,k(s)ds,
from which we deduce
(4.19)CN(t,s)=g(t)(∑(n,k)∈IN(∫U1[0,t](u)f(u)ϕn,k(u)du)(∫U1[0,s](v)f(v)ϕn,k(v)dv)T)g(s)T.
Defining the auxiliary ℝd×d-valued function
(4.20)κn,k(t)=∫U1[0,t](u)f(u)ϕn,k(u)du,
we observe that the (i,j)-coefficient function reads
(4.21)(κn,k)i,j(t)=∫U1[0,t](u)(li(f(u))T⋅cj(ϕn,k(u)))du=∫U(1[0,t](u)ci(fT(u))T⋅cj(ϕn,k(u)))du,
where 1[0,t] is the real function that is one if 0≤u≤t and zero otherwise. As we can write
(4.22)1[0,t](u)ci(fT(u))=fT(u)⋅[0⋮1[0,t](u)⋮0]⟵i,
we see that the function fi,t=1[0,t]ci(fT) belongs to Lf2, so that we can write (κn,k)i,j(t) as a scalar product in the Hilbert space Lf2:
(4.23)(κn,k)i,j(t)=∫Ufi,tT(u)⋅cj(ϕn,k(u))du=(fi,t,cj(ϕn,k)).
We then specify the (i,j)-coefficient of g-1(t)CN(t,s)(g-1(s))T writing
(4.24)∑(n,k)∈IN(κ(t)⋅κ(s)T)i,j=∑(n,k)∈IN∑p=0d-1(fi,t,cj(ϕn,k))(fj,s,cj(ϕn,k)),
and, remembering that the family of functions cj(ϕn,k) forms a complete orthonormal system of Lf2, we can use the Parseval identity, which reads
(4.25)∑(n,k)∈I(κ(t)⋅κ(s)T)i,j=(fi,t,fj,s)=∫U1[0,t](u)ci(fT(u))T⋅1[0,s](u)cj(fT(u))du=∫0t∧s(f⋅fT)i,j(u)du.
Thanks to this relation, we can conclude the evaluation of the covariance since
(4.26)lim⁡N→∞CN(t,s)=g(t)(∫0t∧s(f⋅fT)(u)du)g(s)T=g(t)h(t∧s)g(s)T.We stress the fact that the relation(4.27)C(t,s)=∑(n,k)∈Iψn,k(t)⋅(ψn,k(s))T=Ψ(t)∘ΨT(s)
provides us with a continuous version of the Cholesky decomposition of the covariance kernel C. Indeed, if we chose σn,k as the Cholesky square root of Σn,k, we remark that the operators Φ are triangular in the following sense: consider the chain of nested vectorial spaces {Fn,k}(n,k)∈ℐ(4.28)F0,0⊂F1,0⊂(F2,0⊂F2,1)⋯⊂(Fn,0⊂⋯⊂Fn,2n-1)⋯⊂Ωξ′
with Fn,k=span{fi,j∣0≤i≤n,0≤j≤k}; then, for every (n,k) in ℐ, the operator Ψ transforms the chain {Fn,k}(n,k)∈ℐ into the chain(4.29)E0,0⊂E1,0⊂(E2,0⊂E2,1)⋯⊂(En,0⊂⋯⊂En,2n-1)⋯⊂xΩ′
with ℰn,k=span{Ψi,j∣0≤i≤n,0≤j≤k}.The fact that this covariance is equal to the covariance of the processX, solution of (2.1) implies that we have the following fundamental result.Theorem 4.5.
The processX∞ is equal in law to the initial Gauss-Markov process X used to construct the basis of functions.Remark 4.6.
Our multiresolution representation of the Gauss-Markov processes appears to be the direct consequence of the fact that, because of the Markov property, the Cholesky decomposition of the finite-dimensional covariance admits a simple inductive continuous limit. More generally, triangularization of the kernel operators has been studied in depth [28, 40–42], and it would be interesting to investigate if these results make possible a similar multiresolution approach for non-Markov Gaussian processes. In this regard, we naturally expect to lose the compactness of the supports of a putative basis.Remark 4.7.
We eventually underline the fact that large deviations related to this convergence can be derived through the use of the Baldi and Caramellino good rate function related to the Gaussian pinned processes [43, 44].


## 4.1.1. Finite-Dimensional Approximations

Considering the infinite-dimensional subspacexΩ′ of C0([0,1],ℝd), let us introduce the equivalence relation ~N as(4.1)x∼Ny⟺∀t∈DN,x(t)=y(t).
We can use the functions Ψ to carry through the structure of ~N on the infinite-dimensional space of coefficients ξΩ′:(4.2)ξ∼Nη⟺Ψ(ξ)∼NΨ(η)⟺∀(n,k)∈IN,ξn,k=ηn,k,
which clearly entails that x~Ny if and only if Δ(x)~NΔ(y). We denote the sets of equivalence classes of xΩ′/~N=ΩxN and ξΩ′/~N=ΩξN, which are both clearly isomorphic xΩN=(ℝd)ℐ=ΩξN. For every N>0, we define the finite-dimensional operators ΨN=ixN∘Ψ∘pξN and ΔN=iξN∘Δ∘pxN, with the help of the canonical projections ξpN:Ωξ′→ΩξN, xpN:Ωx′→ΩxN and the inclusion map iξN:ΩξN→Ωξ′, ixN:ΩxN→Ωx′.The results of the preceding sections straightforwardly extend on the equivalence classes, and in particular we see that the functionsΨN:ΩξN→ΩxN and ΔN:ΩxN→ΩξN are linear finite-dimensional bijections satisfying ΨN=ΔN-1. We write e={ep,q}(p,q)∈ℐ (resp., f={fp,q}(p,q)∈ℐ), the canonical basis of ξΩN (resp., ΩxN) when listed in the recursive dyadic order. In these bases, the matrices ΨN and ΔN are lower block triangular. Indeed, denoting ΨN in the natural bases e={ep,q}(p,q)∈ℐ and f={fp,q}(p,q)∈ℐ by(4.3)ΨN=[ψn,k(mi,j)]=[Ψn,ki,j],
where Ψn,ki,j is a d×d matrix, the structure of the nested support Sn,k entails the block-triangular structure (where only possibly nonzero coefficients are written):(4.4)ΨαN=[ψ0,00,0ψ0,01,0ψ1,01,0ψ0,02,0ψ1,02,0ψ2,02,0ψ0,02,1ψ1,02,1ψ2,12,1ψ0,03,0ψ1,03,0ψ2,03,0ψ3,03,0ψ0,03,1ψ1,03,1ψ2,03,1ψ3,13,1ψ0,03,2ψ1,03,2ψ2,13,2ψ3,23,2ψ0,03,3ψ1,03,3ψ2,13,3ψ3,33,3⋮⋱].Similarly, for the matrix representation ofΔN in the natural bases en,k and fi,j(4.5)ΔN=[Δi,jn,k]
proves to have the following triangular form:(4.6)ΔN=[g-1(t0,0)TM0,0-g-1(t0,0)TR1,0g-1(t1,0)TM1,0-g-1(t0,0)TR2,0g-1(t1,0)TM2,0-g-1(t0,0)TR2,1-g-1(t1,0)TL2,1g-1(t2,1)TM2,1⋮⋱].
The duality property, Proposition 3.8, simply reads for all 0≤n<N and 0≤k<2n-1, 0≤p<N and 0≤p<2q-1(4.7)P(δp,q,ψn,k)=∑(n,k)∈INΔi,jp,q⋅Ψn,ki,j=δn,kp,qId,
that is, ΔN·ΨN=IdξΩN.  But because  we are now in a finite-dimensional setting, we also have ΨN·ΔN=IdxΩN:(4.8)δk,li,jId=∑(p,q)∈INΨp,qi,j⋅Δk,lp,q.
Realizing that δk,li,jId represents the class of functions x in xΩ′ whose values are zero on every dyadic point  of DN except for x(l2k)=Id, {Δk,lp,q}(p,q)∈ℐN clearly appear as the coefficients of the decomposition of such functions in the bases ψp,q for (p,q) in ℐN.DenotingΞ={Ξn,k}(n,k)∈I, a set of independent Gaussian variables of law 𝒩(0,Id) on (Ω,ℱ,P), and for all N>0, we form the finite dimensional Gauss-Markov vector [Xi,jN](i,j)∈ℐN as(4.9)Xi,jN=∑(n,k)∈INψn,k(mi,j)⋅Ξn,k,
which, from Corollary 2.5, has the same law as [Xt]t∈DN, the finite-dimensional random vector obtained from sampling X on DN (modulo a permutation on the indices). We then prove the following lemma that sheds light on the meaning of the construction.Lemma 4.1.
The Cholesky decomposition of the finite-dimensional covariance block matrixΣN is given by ΣN=ΨN·ΨNT.Proof.
For every0≤t,s≤1, we compute the covariance of the finite-dimensional process XN as
(4.10)CN(t,s)=E[XtN⋅(XsN)T]=∑n=0N∑0≤k<2n-1ψn,k(t)⋅(ψn,k(s))T.
From there, we write the finite-dimensional covariance block matrix ΣN in the recursively ordered basis fi,j for 0≤i≤N, 0≤j<2i-1, as
(4.11)[𝚺N]k,li,j=CN(mi,j,mk,l)=∑n=0N∑0≤k<2n-1Ψn,ki,j⋅ψn,kk,l.
We already established that the matrix ΨN was triangular with positive diagonal coefficient, which entails that the preceding equality provides us with the Cholesky decomposition of Σ.In the finite-dimensional case, the inverse covariance or potential matrix is a well-defined quantity and we straightforwardly have the following corollary.Corollary 4.2.
The Cholesky decomposition of the finite-dimensional inverse covariance matrixΣN-1 is given by ΣN-1=ΔNT·ΔN.Proof.
The result stems for the equalitiesΣN-1=(ΨN·ΨNT)-1=(ΨN-1)T·ΨN-1=ΔNT·ΔN.


## 4.1.2. The Lévy-Ciesielski Expansion

We now show that, asymptotically, the basesψn,k allow  us to faithfully build the Gauss-Markov process from which we have derived its expression. In this perspective we consider Ξ={Ξn,k}(n,k)∈ℐ, a set of independent Gaussian variables of law 𝒩(0,Id) on (Ω,ℱ,P), and, for all N>0, we form the finite-dimensional continuous Gaussian process ZN, defined for 0≤t≤1 by(4.12)XtN=∑(n,k)∈INψn,k(t)⋅Ξn,k,
which, from the result of Theorem 2.4, has the same law ZtN=𝔼[Xt∣ℱN]. We prove the following lemma.Lemma 4.3.
The sequence of processesXN almost surely converges towards a continuous Gaussian process denoted by X∞.Proof.
For all fixedN>0 and for any ω in Ω, we know that t↦XtN(ω) is continuous. Moreover, we have established, that, for every ξ in ξΩ′, XN(ξ) converges uniformly in t toward a continuous limit denoted by XN(ξ). Therefore, in order to prove that limN→∞XN defines almost surely a process X with continuous paths, it is sufficient to show that Pξ(ξΩ′)=1, where Pξ=PΞ-1 is the Ξ-induced measure on ξΩ, which stems from a classical Borel-Cantelli argument. For ξ a random variable of normal law 𝒩(0,1), and a>0, we have
(4.13)P(|ξ|>a)=2π∫a∞e-u2/2du≤2π∫a∞uae-u2/2du=2πe-a2/2a.
Then, for any δ>0(4.14)Pξ(max⁡0≤k<2n-1|ξn,k|∞>2nδ/2)≤d2nP(|ξ|>d2nδ/2)=2π2(1-δ/2)nexp⁡(-2nδ-1).
Since the series
(4.15)∑n=0∞2π2(1-δ/2)nexp⁡(-2nδ-1)
is convergent, the Borel-Cantelli argument implies that Pξ(ξΩ′)=1. Eventually, the continuous almost-sure limit process Xt∞ is Gaussian as a countable sum of Gaussian processes.Now that these preliminary remarks have been made, we can evaluate, for anyt and s in [0,1], the covariance of X as the limit of the covariance of XN.Lemma 4.4.
For any0≤t,s≤1, the covariance of X∞={Xt∞=Ψt∘Ξ;0≤t≤1} is
(4.16)C(t,s)=E[Xt∞⋅(Xs∞)T]=g(t)h(t∧s)g(s)T.Proof.
AsΞn,k are independent Gaussian random variables of normal law 𝒩(0,Id), we see that the covariance of XN is given by
(4.17)CN(t,s)=E[XtN⋅(XsN)T]=∑(n,k)∈INψn,k(t)⋅(ψn,k(s))T.
To compute the limit of the right-hand side, we need to remember that the element of the bases ψn,k and the functions ϕn,k are linked by the following relation:
(4.18)ψn,k(t)=K[ϕn,k]=g(t)∫U1[0,t](s)f(s)ϕn,k(s)ds,
from which we deduce
(4.19)CN(t,s)=g(t)(∑(n,k)∈IN(∫U1[0,t](u)f(u)ϕn,k(u)du)(∫U1[0,s](v)f(v)ϕn,k(v)dv)T)g(s)T.
Defining the auxiliary ℝd×d-valued function
(4.20)κn,k(t)=∫U1[0,t](u)f(u)ϕn,k(u)du,
we observe that the (i,j)-coefficient function reads
(4.21)(κn,k)i,j(t)=∫U1[0,t](u)(li(f(u))T⋅cj(ϕn,k(u)))du=∫U(1[0,t](u)ci(fT(u))T⋅cj(ϕn,k(u)))du,
where 1[0,t] is the real function that is one if 0≤u≤t and zero otherwise. As we can write
(4.22)1[0,t](u)ci(fT(u))=fT(u)⋅[0⋮1[0,t](u)⋮0]⟵i,
we see that the function fi,t=1[0,t]ci(fT) belongs to Lf2, so that we can write (κn,k)i,j(t) as a scalar product in the Hilbert space Lf2:
(4.23)(κn,k)i,j(t)=∫Ufi,tT(u)⋅cj(ϕn,k(u))du=(fi,t,cj(ϕn,k)).
We then specify the (i,j)-coefficient of g-1(t)CN(t,s)(g-1(s))T writing
(4.24)∑(n,k)∈IN(κ(t)⋅κ(s)T)i,j=∑(n,k)∈IN∑p=0d-1(fi,t,cj(ϕn,k))(fj,s,cj(ϕn,k)),
and, remembering that the family of functions cj(ϕn,k) forms a complete orthonormal system of Lf2, we can use the Parseval identity, which reads
(4.25)∑(n,k)∈I(κ(t)⋅κ(s)T)i,j=(fi,t,fj,s)=∫U1[0,t](u)ci(fT(u))T⋅1[0,s](u)cj(fT(u))du=∫0t∧s(f⋅fT)i,j(u)du.
Thanks to this relation, we can conclude the evaluation of the covariance since
(4.26)lim⁡N→∞CN(t,s)=g(t)(∫0t∧s(f⋅fT)(u)du)g(s)T=g(t)h(t∧s)g(s)T.We stress the fact that the relation(4.27)C(t,s)=∑(n,k)∈Iψn,k(t)⋅(ψn,k(s))T=Ψ(t)∘ΨT(s)
provides us with a continuous version of the Cholesky decomposition of the covariance kernel C. Indeed, if we chose σn,k as the Cholesky square root of Σn,k, we remark that the operators Φ are triangular in the following sense: consider the chain of nested vectorial spaces {Fn,k}(n,k)∈ℐ(4.28)F0,0⊂F1,0⊂(F2,0⊂F2,1)⋯⊂(Fn,0⊂⋯⊂Fn,2n-1)⋯⊂Ωξ′
with Fn,k=span{fi,j∣0≤i≤n,0≤j≤k}; then, for every (n,k) in ℐ, the operator Ψ transforms the chain {Fn,k}(n,k)∈ℐ into the chain(4.29)E0,0⊂E1,0⊂(E2,0⊂E2,1)⋯⊂(En,0⊂⋯⊂En,2n-1)⋯⊂xΩ′
with ℰn,k=span{Ψi,j∣0≤i≤n,0≤j≤k}.The fact that this covariance is equal to the covariance of the processX, solution of (2.1) implies that we have the following fundamental result.Theorem 4.5.
The processX∞ is equal in law to the initial Gauss-Markov process X used to construct the basis of functions.Remark 4.6.
Our multiresolution representation of the Gauss-Markov processes appears to be the direct consequence of the fact that, because of the Markov property, the Cholesky decomposition of the finite-dimensional covariance admits a simple inductive continuous limit. More generally, triangularization of the kernel operators has been studied in depth [28, 40–42], and it would be interesting to investigate if these results make possible a similar multiresolution approach for non-Markov Gaussian processes. In this regard, we naturally expect to lose the compactness of the supports of a putative basis.Remark 4.7.
We eventually underline the fact that large deviations related to this convergence can be derived through the use of the Baldi and Caramellino good rate function related to the Gaussian pinned processes [43, 44].


## 4.2. Optimality Criterion of the Decomposition

In the following, we draw from the theory of interpolating splines to further characterize the nature of our proposed basis for the construction of the Gauss-Markov processes. Essentially adapting the results from the previous works [45, 46], we first show that the finite-dimensional sample paths of our construction induce a nested sequence ℰN of the reproducing Hilbert kernel space (RKHS). In turn, the finite-dimensional process XN naturally appears as the orthogonal projection of the infinite-dimensional process X onto ℰN. We then show that such an RKHS structure allows us to define a unicity criterion for the finite-dimensional sample path as the only functions of ℰ that minimize a functional, called Dirichlet energy, under constraint of interpolation on DN (equivalent to conditioning on the times DN). In this respect, we point out that the close relation between the Markov processes and the Dirichlet forms is the subject of a vast literature, largely beyond the scope of the present paper (see, e.g., [18]).


### 4.2.1. Sample Paths Space as a Reproducing Hilbert Kernel Space

In order to define the finite-dimensional sample paths as a nested sequence of RKHSs, let us first define the infinite-dimensional operator(4.30)Φ:{l2(ξΩ)⟼Lf2,ξ⟼Φ[ξ]={t⟼∑(n,k)∈Iϕn,k(t)⋅ξn,k}.
Since we know that the column functions of ϕn,k form a complete orthonormal system of Lf2, the operator Φ is an isometry and its inverse satisfies Φ-1=ΦT, which reads for all v in Lf2(4.31)[Φ-1[v]]n,k=∫Uϕn,kT(t)⋅v(t)dt=P(ϕn,k,v).
Equipped with this infinite-dimensional isometry, we then consider the linear operator ℒ=Φ∘Δ suitably defined on the set(4.32)E={u∈C0(U,Rd)∣L[u]∈Lf2}={u∈C0(U,Rd)∣Δ[u]∈l2(ξΩ)}
with ∥ξ∥22=∑n,k∈ℐ|ξn,k|22, the l2 norm of ξΩ. The set ℰ form an infinite-dimensional vectorial space that is naturally equipped with the inner product(4.33)∀(u,v)∈E2,〈u,v〉=∫UL[u](t)T⋅L[v](t)dt=(L[u],L[v]).
Moreover since u(0)=v(0)=0, such an inner product is definite positive and,  consequently, ℰ forms an Hilbert space.Remark 4.8.
Two straightforward remarks are worth making. First, the spaceℰ is strictly included in the infinite-dimensional sample paths space xΩ′. Second, notice that, in the favorable case m=d, if f is everywhere invertible with continuously differentiable inverse, we have ℒ=𝒟=𝒦-1. More relevantly, the operator ℒ can actually consider a first-order differential operator from ℰ to Lf2 as a general left inverse of the integral operator 𝒦. Indeed, realizing that on Lf2,𝒦 can be expressed as 𝒦=Ψ∘Φ-1, we clearly have
(4.34)L∘K=Φ∘Δ∘Ψ∘Φ-1=IdLf2.We know motivate the introduction of the Hilbert spaceℰ by the following claim.Proposition 4.9.
The Hilbert space(ℰ,〈,〉) is a reproducing kernel Hilbert space (RKHS) with ℝd×d-valued reproducing kernel C, the covariance function of the process X.Proof.
Consider the problem of finding all elementsu of ℰ solution of the equation ℒ[u]=v for v in Lf2. The operator 𝒦 provides us with a continuous ℝd×m-valued kernel function k:
(4.35)∀(t,s)∈U2,k(t,s)=1[0,t](s)g(t)⋅f(s),
which is clearly the Green function for our differential equation. This entails that the following equalitiy holds for every u in ℰ:
(4.36)u(t)=∫Uk(t,s)L[u](s)ds.
Moreover, we can decompose the kernel k in the Lf2 sense as
(4.37)k(t,s)=∑(n,k)∈Iψn,k(t)⋅ϕn,kT(s)
since we have
(4.38)k(t,s)=K[δsIdLf2](t)=K[∑(n,k)∈Iϕn,k⋅ϕn,kT(s)](t)=∑(n,k)∈IK[ϕn,k](t)⋅ϕn,kT(s),
with  δs=δ(·-s). Then, we clearly have
(4.39)C(t,s)=∫Uk(t,u)⋅k(s,u)Tdu=∑(n,k)∈Iψn,k(t)⋅ψn,kT(s),
where we recognize the covariance function of X, which implies
(4.40)k(t,s)=∑(n,k)∈Iψn,k(t)⋅L[ϕn,k]T(s)=L[C(t,⋅)].
Eventually, for all u in Lf2, we have
(4.41)u(t)=∫UL[C(t,⋅)](s)⋅L[u](s)ds=P〈C(t,⋅),u〉,
where we have introduced the 𝒫-operator associated with the inner product 〈,〉: for all ℝd×d-valued functions A and B defined on U such that the columns ci(A) and ci(B), 0≤i<d, are in ℰ, we define the matrix 𝒫〈A,B〉 in ℝd×d by
(4.42)∀0≤i,j<d,P〈A,B〉i,j=〈ci(A),cj(B)〉.
By the Moore-Aronszajn theorem [47], we deduce that there is a unique reproducing kernel Hilbert space associated with a given covariance kernel. Thus, ℰ is the reproducing subspace of C0(U,ℝd) corresponding to the kernel C, with respect to the inner product 〈,〉.Remark 4.10.
From a more abstract point of view, it is well known that the covariance operator of a Gaussian measure defines an associated Hilbert structure [48, 49].In the sequel, we will use the spaceℰ as the ambient Hilbert space to define the finite-dimensional sample paths spaces as a nested sequence of RKHS. More precisely, let us write for ℰN the finite-dimensional subspace of ℰ(4.43)EN={u∈C0(U,Rd)∣L[u]∈Lf,N2},
with the space Lf,N2 being defined as(4.44)Lf,N2=span⁡[{ci(ϕn,k)}n,k∈IN,0≤i<d].We refer to such spaces as finite-dimensional approximation spaces  since we remark that(4.45)EN=span⁡[{ci(ψn,k)}n,k∈IN,0≤i<d]=ΨN[ξΩN],
which means that the space ℰN is made of the sample space of the finite-dimensional process XN. The previous definition makes obvious the nested structure ℰ0⊂ℰ1⊂⋯⊂ℰ, and it is easy to characterize each space ℰN as a reproducing Hilbert kernel space.Proposition 4.11.
The Hilbert spaces(ℰN,〈,〉) are reproducing kernel Hilbert spaces (RKHSs) with ℝd×d-valued reproducing kernel CN, the covariance function of the process XN.Proof.
The proof this proposition follows the exact same argument as that in the case ofℰ, but with the introduction of finite-dimensional kernels kN(4.46)∀(t,s)∈U2,kN(t,s)=∑(n,k)∈INψn,k(t)⋅ϕn,kT(s),
and the corresponding covariance function
(4.47)∀(t,s)∈[0,1]2,CN(t,s)=∑(n,k)∈INψn,k(t)⋅ψn,kT(s).


### 4.2.2. Finite-Dimensional Processes as Orthogonal Projections

The framework set in the previous section offers a new interpretation of our construction. Indeed, for allN>0, the columns of {ψn,k}(n,k)∈ℐN form an orthonormal basis of the space ℰN:(4.48)P〈ψn,k,ψp,q〉=P(L[ψn,k],L[ψp,q])=P(ϕn,k,ϕp,q)=δp,qn,k.
This leads to defining the finite-dimensional approximation xN of an sample path x of ℰ as the orthogonal projection of x on ℰN with respect to the inner product 〈,〉. At this point, it is worth remembering that the space ℰ is strictly contained in xΩ′ and does not coincide with xΩ′: actually one can easily show that P(ℰ)=0. We devote the rest of this section to defining the finite-dimensional processes ZN=𝔼N[X] resulting from the conditioning on DN, as pathwise orthogonal projection of the original process X on the sample space ℰN.Proposition 4.12.
For anyN>0, the conditioned processes 𝔼N[X] can be written as the orthogonal projection of X on ℰN with respect to 〈,〉:
(4.49)EN[X]=∑(n,k)∈INψn,k⋅P〈ψn,k,X〉.The only hurdle to prove Proposition4.12 is purely technical in the sense that the process X exists in a larger space than ℰ: we need to find a way to extend the definition of 〈,〉 so that the expression bears a meaning. Before answering this point quite straightforwardly, we need to establish the following lemma.Lemma 4.13.
Writing the Gauss-Markov processXt=∫01k(t,s)dWs, for all N>0, the conditioned process ZN=𝔼N[X] is expressed as the stochastic integral
(4.50)ZN=∫01kN(t,s)dWswithkN(t,s)=∑(n,k)∈INψn,k(t)⋅ϕn,kT(s).Proof.
In the previous section, we have noticed that the kernelkN converges toward the kernel k (the Green function) in the Lf2 sense
(4.51)k(t,s)=∑(n,k)∈Iψn,k(t)⋅ϕn,kT(s)=lim⁡N→∞∑(n,k)∈INψn,k(t)⋅ϕn,kT(s)kN=lim⁡N→∞kN(t,s).
This implies that the process X as the stochastic integral  can also be written as
(4.52)Xt=g(t)∫U1[0,t](s)(s)dWs=∫01k(t,s)dWs=lim⁡N→∞∫01kN(t,s)dWs.
Specifying the decomposition of kN, we can then naturally express X as the convergent sum
(4.53)Xt=∑(n,k)∈Iψn,k⋅Ξn,kwithΞn,k=∫01ϕn,kT(s)dWs,
where the orthonormality property of the ϕn,k with respect to (,) makes the vectors Ξn,k appear as independent d-dimensional Gaussian variables of law 𝒩(0,Id). It is then easy to see that by definition of the elements ψn,k, for almost every ω in Ω, we then have
(4.54)∀N>0,0≤t≤1,ZN(ω)=EN[X](ω)=∑(n,k)∈INψn,k⋅Ξn,k(ω),
and we finally recognize in the previous expression that for all 0≤t≤1(4.55)ZtN=∑(n,k)∈INψn,k⋅Ξn,k=∑(n,k)∈INψn,k(t)⋅∫Uϕn,kT(s)dWs=∫01kN(t,s)dWs.We can now proceed to justify the main result of Proposition4.12.Proof.
The finite-dimensional processesZN defined through Lemma 4.13 have sample paths t↦ZtN(ω) belonging to ℰN. Moreover, for almost every ω in Ω  and for all n,k in ℐN,
(4.56)P〈ψn,k,ZN(ω)〉=P〈ψn,k,∫01kN(t,s)dWs(ω)〉=P〈ψn,k,∑(p,q)∈INψp,q(ω)⋅∫01ϕp,qT(s)dWs(ω)〉=∫01ϕn,kT(s)dWs(ω),
because of the orthonormality property of ψn,k with respect to 〈,〉. As the previous equalities hold  for every N>0, the applications x↦𝒫〈ψn,k,x〉 can naturally be extended on xΩ′ by continuity. Therefore, it makes sense to write, for all (n,k) in ℐN, 𝒫〈ψn,k,ZN〉=limN→∞𝒫〈ψn,k,ZN〉=def𝒫〈ψn,k,X〉 even if   X is defined into a larger sample space than ℰ. In other words, we have
(4.57)P〈ψn,k,X〉=∫01ϕn,kT(s)dWs=Ξn,k,
and we can thus express the conditioned process ZN=𝔼N[X] as the orthogonal projection of X onto the finite sample path ℰN by writing
(4.58)ZN=∑(n,k)∈INψn,k⋅P〈ψn,k,X〉.


### 4.2.3. Optimality Criterion of the Sample Paths

Proposition4.12 elucidates the structure of the conditioned processes ZN as pathwise orthogonal projections of X on the finite-dimensional RKHS ℰN. It allows us to cast the finite sample paths in a geometric setting and incidentally, to give a characterization of them as the minimizer of some functionals. In doing so, we shed a new light on well-known results of the interpolation theory [50–52] and extend them to the multidimensional case.The central point of this section reads as follows.Proposition 4.14.
Given a functionx in ℰ, the function xN=(Ψ∘ΔN)[x] belongs to ℰN and is defined by the following optimal criterion: xN is the only function in ℰ interpolating x on DN such that the functional
(4.59)〈y,y〉=‖L[y](t)‖22=∫01|L[y](t)|22dt
takes its unique minimal value over ℰ in xN.Proof.
The spaceℰN has been defined as ℰN=ΨN[ξΩN]=Ψ∘ΔN[ℰ] so that, for all x in ℰ, xN clearly belongs to ℰN. Moreover, xN interpolates x on DN: indeed, we know that the finite-dimensional operators ΔN and ΨN-1 are inverse of each other ΔN=ΨN-1, which entails that for all t in DN(4.60)xN(t)=(Ψ∘ΔN)[x](t)=(ΨN∘ΔN)[x](t)=x(t),
where we use the fact that, for any ξ in ξΩ′  and for all t in DN, ΨN[ξ](t)=Ψ[ξ](t) (recall that ψn,k(t)=0 if n>N and t belongs to DN).
Let us now show thatxN is determined in ℰ by the announced optimal criterion. Suppose y belongs to ℰ and interpolates x on DN, and remark that we can write
(4.61)〈y,y〉=‖L[y]‖22=‖(Φ∘Δ)[y](t)‖22=‖Δ[y]‖22,
since Φ is an isometry. Then, consider Δ[y] in l2(ξΩ) and remark that, since for all (n,k) in ℐN, δn,k are Dirac measures supported by DN, we have
(4.62)∀(n,k)∈IN,Δn,k[y]=P(δn,k,y)=P(δn,k,x)=Δn,k[x]=Δn,k[xN].
This entails
(4.63)‖Δ[y]‖22dt=∑(n,k)∈I|Δn,k[y]|22≥∑(n,k)∈IN|Δn,k[y]|22=‖Δ[xN]‖22dt.
Since, by definition of xN, δn,k[xN]=0 if n>N. Moreover, the minimum 〈xN,xN〉 is only attained for y such that δn,k[y]=0 if n>N and δn,k[y]=δn,k[x] if n≤N, which defines univocally xN. This establishes that, for all y in ℰ such that for all t in DN, y(t)=x(t), we have 〈xN,xN〉≤〈y,y〉 and the equality case holds if and only if y=xN.Remark 4.15.
Whenℒ represents a regular differential operator of order d, ∑i=1dai(t)Di, where D=d/dt, that is, for
(4.64)dXt=α(t)⋅Xt+Γ(t)⋅dWt,
with
(4.65)α(t)=[01⋱⋱⋱1adad-1…a1],Γ(t)=[00⋮1].
The finite-dimensional sample paths coincide exactly with the spline interpolation of order 2d+1, which are well known to satisfy the previous criterion [46]. This example will be further explored in the example section.The Dirichlet energy simply appears as the squared norm induced onℰ by the inner product 〈,〉, which in turn can be characterized as a Dirichlet quadratic form on ℰ. Actually, such a Dirichlet form can be used to define the Gauss-Markov process, extending the Gauss-Markov property to processes indexed on the multidimensional spaces parameter [19]. In particular, for an n-dimensional parameter space, we can condition such Gauss-Markov processes on a smooth n-1-dimensional boundary. Within the boundary, the sample paths of the resulting conditioned process (the solution to the prediction problem in [19]) are the solutions to the corresponding Dirichlet problems for the elliptic operator associated with the Dirichlet form.The characterization of the basis as the minimizer of such a Dirichlet energy (4.59) gives rise to an alternative method to compute the basis as the solution of a Dirichlet boundary value problem for an elliptic differential operator.Proposition 4.16.
Let us assume thatα and Γ are continuously differentiable and that Γ is invertible. Then, the functions μn,k are defined as
(4.66)μn,k(t)={μl(t),t∈[ln,k,mn,k],μr(t),t∈[mn,k,rn,k],0,else,
where μl and μr are the unique solutions of the second-order d-dimensional linear differential equation
(4.67)u′′+(Γ-1(αTΓ-Γ′)-α)u′-(Γ-1(αTΓ-Γ′)α+α′)u=0
with the following boundary value conditions:
(4.68)μl(ln,k)=0,μl(mn,k)=Id,μr(mn,k)=Id,μr(rn,k)=0.Proof.
By Proposition4.14, we know that μn,k(t) minimizes the convex functional
(4.69)∫01|L[u](s)|22ds
over ℰ, being equal to zero outside the interval [ln,k,rn,k] and equal to one at the point t=mn,k. Because of the hypotheses on α and Γ, we have ℒ=𝒟 and we can additionally restrain our search to functions that are twice continuously differentiable. Incidentally, we only need to minimize separately the contributions on the interval [ln,k,mn,k] and [mn,k,rn,k]. On both intervals, this problem is a classical Euler-Lagrange problem (see, e.g., [53]) and is solved using basic principles of calculus of variations. We easily identify the Lagrangian of our problem as
(4.70)L(t,u,u′)=|(u′-α(t)u(t))(Γ(t))-1|22=(u′(t)-α(t)u(t))T(Γ(t))-1(u′(t)-α(t)u(t)).
From there, after some simple matrix calculations, the Euler-Lagrange equations
(4.71)∂L(t,u,u′)∂ui-ddt(∂L(t,u,u′)∂u′i)=0,i=1,…,d,
can be expressed under the form:
(4.72)u′′+(Γ-1(αTΓ-Γ′)-α)u′-(Γ-1(αTΓ-Γ′)α+α′)u=0,
which ends the proof.Remark 4.17.
It is a simple matter of calculus to check that the expression ofμ given in Proposition 2.1 satisfies (4.67). Notice also that, in the case Γ=Id, the differential equation becomes
(4.73)u′′+(αT-α)u′-(αTα+α′)u=0,
which is further simplified for constant α.Under the hypotheses of Proposition4.16, we can thus define μn,k as the unique solution to the second-order linear differential equation (4.67) with the appropriate boundary values conditions. From this definition, it is then easy to derive the bases ψn,k by completing the following program.(1)
Compute thet↦μn,k(t) by solving the linear ordinary differential problem.(2)
Apply the differential operator𝒟 to get the functions 𝒟[μn,k].(3)
Orthonormalize the column functionst↦cj(𝒟[μn,k(t)]) by the Gram-Schmidt process.(4)
Apply the integral operator𝒦 to get the desired functions ψn,k (or equivalently multiply the original function t↦μn,k(t) by the corresponding Gram-Schmidt triangular matrix).Notice finally that each of these points is easily implemented numerically.


## 4.2.1. Sample Paths Space as a Reproducing Hilbert Kernel Space

In order to define the finite-dimensional sample paths as a nested sequence of RKHSs, let us first define the infinite-dimensional operator(4.30)Φ:{l2(ξΩ)⟼Lf2,ξ⟼Φ[ξ]={t⟼∑(n,k)∈Iϕn,k(t)⋅ξn,k}.
Since we know that the column functions of ϕn,k form a complete orthonormal system of Lf2, the operator Φ is an isometry and its inverse satisfies Φ-1=ΦT, which reads for all v in Lf2(4.31)[Φ-1[v]]n,k=∫Uϕn,kT(t)⋅v(t)dt=P(ϕn,k,v).
Equipped with this infinite-dimensional isometry, we then consider the linear operator ℒ=Φ∘Δ suitably defined on the set(4.32)E={u∈C0(U,Rd)∣L[u]∈Lf2}={u∈C0(U,Rd)∣Δ[u]∈l2(ξΩ)}
with ∥ξ∥22=∑n,k∈ℐ|ξn,k|22, the l2 norm of ξΩ. The set ℰ form an infinite-dimensional vectorial space that is naturally equipped with the inner product(4.33)∀(u,v)∈E2,〈u,v〉=∫UL[u](t)T⋅L[v](t)dt=(L[u],L[v]).
Moreover since u(0)=v(0)=0, such an inner product is definite positive and,  consequently, ℰ forms an Hilbert space.Remark 4.8.
Two straightforward remarks are worth making. First, the spaceℰ is strictly included in the infinite-dimensional sample paths space xΩ′. Second, notice that, in the favorable case m=d, if f is everywhere invertible with continuously differentiable inverse, we have ℒ=𝒟=𝒦-1. More relevantly, the operator ℒ can actually consider a first-order differential operator from ℰ to Lf2 as a general left inverse of the integral operator 𝒦. Indeed, realizing that on Lf2,𝒦 can be expressed as 𝒦=Ψ∘Φ-1, we clearly have
(4.34)L∘K=Φ∘Δ∘Ψ∘Φ-1=IdLf2.We know motivate the introduction of the Hilbert spaceℰ by the following claim.Proposition 4.9.
The Hilbert space(ℰ,〈,〉) is a reproducing kernel Hilbert space (RKHS) with ℝd×d-valued reproducing kernel C, the covariance function of the process X.Proof.
Consider the problem of finding all elementsu of ℰ solution of the equation ℒ[u]=v for v in Lf2. The operator 𝒦 provides us with a continuous ℝd×m-valued kernel function k:
(4.35)∀(t,s)∈U2,k(t,s)=1[0,t](s)g(t)⋅f(s),
which is clearly the Green function for our differential equation. This entails that the following equalitiy holds for every u in ℰ:
(4.36)u(t)=∫Uk(t,s)L[u](s)ds.
Moreover, we can decompose the kernel k in the Lf2 sense as
(4.37)k(t,s)=∑(n,k)∈Iψn,k(t)⋅ϕn,kT(s)
since we have
(4.38)k(t,s)=K[δsIdLf2](t)=K[∑(n,k)∈Iϕn,k⋅ϕn,kT(s)](t)=∑(n,k)∈IK[ϕn,k](t)⋅ϕn,kT(s),
with  δs=δ(·-s). Then, we clearly have
(4.39)C(t,s)=∫Uk(t,u)⋅k(s,u)Tdu=∑(n,k)∈Iψn,k(t)⋅ψn,kT(s),
where we recognize the covariance function of X, which implies
(4.40)k(t,s)=∑(n,k)∈Iψn,k(t)⋅L[ϕn,k]T(s)=L[C(t,⋅)].
Eventually, for all u in Lf2, we have
(4.41)u(t)=∫UL[C(t,⋅)](s)⋅L[u](s)ds=P〈C(t,⋅),u〉,
where we have introduced the 𝒫-operator associated with the inner product 〈,〉: for all ℝd×d-valued functions A and B defined on U such that the columns ci(A) and ci(B), 0≤i<d, are in ℰ, we define the matrix 𝒫〈A,B〉 in ℝd×d by
(4.42)∀0≤i,j<d,P〈A,B〉i,j=〈ci(A),cj(B)〉.
By the Moore-Aronszajn theorem [47], we deduce that there is a unique reproducing kernel Hilbert space associated with a given covariance kernel. Thus, ℰ is the reproducing subspace of C0(U,ℝd) corresponding to the kernel C, with respect to the inner product 〈,〉.Remark 4.10.
From a more abstract point of view, it is well known that the covariance operator of a Gaussian measure defines an associated Hilbert structure [48, 49].In the sequel, we will use the spaceℰ as the ambient Hilbert space to define the finite-dimensional sample paths spaces as a nested sequence of RKHS. More precisely, let us write for ℰN the finite-dimensional subspace of ℰ(4.43)EN={u∈C0(U,Rd)∣L[u]∈Lf,N2},
with the space Lf,N2 being defined as(4.44)Lf,N2=span⁡[{ci(ϕn,k)}n,k∈IN,0≤i<d].We refer to such spaces as finite-dimensional approximation spaces  since we remark that(4.45)EN=span⁡[{ci(ψn,k)}n,k∈IN,0≤i<d]=ΨN[ξΩN],
which means that the space ℰN is made of the sample space of the finite-dimensional process XN. The previous definition makes obvious the nested structure ℰ0⊂ℰ1⊂⋯⊂ℰ, and it is easy to characterize each space ℰN as a reproducing Hilbert kernel space.Proposition 4.11.
The Hilbert spaces(ℰN,〈,〉) are reproducing kernel Hilbert spaces (RKHSs) with ℝd×d-valued reproducing kernel CN, the covariance function of the process XN.Proof.
The proof this proposition follows the exact same argument as that in the case ofℰ, but with the introduction of finite-dimensional kernels kN(4.46)∀(t,s)∈U2,kN(t,s)=∑(n,k)∈INψn,k(t)⋅ϕn,kT(s),
and the corresponding covariance function
(4.47)∀(t,s)∈[0,1]2,CN(t,s)=∑(n,k)∈INψn,k(t)⋅ψn,kT(s).


## 4.2.2. Finite-Dimensional Processes as Orthogonal Projections

The framework set in the previous section offers a new interpretation of our construction. Indeed, for allN>0, the columns of {ψn,k}(n,k)∈ℐN form an orthonormal basis of the space ℰN:(4.48)P〈ψn,k,ψp,q〉=P(L[ψn,k],L[ψp,q])=P(ϕn,k,ϕp,q)=δp,qn,k.
This leads to defining the finite-dimensional approximation xN of an sample path x of ℰ as the orthogonal projection of x on ℰN with respect to the inner product 〈,〉. At this point, it is worth remembering that the space ℰ is strictly contained in xΩ′ and does not coincide with xΩ′: actually one can easily show that P(ℰ)=0. We devote the rest of this section to defining the finite-dimensional processes ZN=𝔼N[X] resulting from the conditioning on DN, as pathwise orthogonal projection of the original process X on the sample space ℰN.Proposition 4.12.
For anyN>0, the conditioned processes 𝔼N[X] can be written as the orthogonal projection of X on ℰN with respect to 〈,〉:
(4.49)EN[X]=∑(n,k)∈INψn,k⋅P〈ψn,k,X〉.The only hurdle to prove Proposition4.12 is purely technical in the sense that the process X exists in a larger space than ℰ: we need to find a way to extend the definition of 〈,〉 so that the expression bears a meaning. Before answering this point quite straightforwardly, we need to establish the following lemma.Lemma 4.13.
Writing the Gauss-Markov processXt=∫01k(t,s)dWs, for all N>0, the conditioned process ZN=𝔼N[X] is expressed as the stochastic integral
(4.50)ZN=∫01kN(t,s)dWswithkN(t,s)=∑(n,k)∈INψn,k(t)⋅ϕn,kT(s).Proof.
In the previous section, we have noticed that the kernelkN converges toward the kernel k (the Green function) in the Lf2 sense
(4.51)k(t,s)=∑(n,k)∈Iψn,k(t)⋅ϕn,kT(s)=lim⁡N→∞∑(n,k)∈INψn,k(t)⋅ϕn,kT(s)kN=lim⁡N→∞kN(t,s).
This implies that the process X as the stochastic integral  can also be written as
(4.52)Xt=g(t)∫U1[0,t](s)(s)dWs=∫01k(t,s)dWs=lim⁡N→∞∫01kN(t,s)dWs.
Specifying the decomposition of kN, we can then naturally express X as the convergent sum
(4.53)Xt=∑(n,k)∈Iψn,k⋅Ξn,kwithΞn,k=∫01ϕn,kT(s)dWs,
where the orthonormality property of the ϕn,k with respect to (,) makes the vectors Ξn,k appear as independent d-dimensional Gaussian variables of law 𝒩(0,Id). It is then easy to see that by definition of the elements ψn,k, for almost every ω in Ω, we then have
(4.54)∀N>0,0≤t≤1,ZN(ω)=EN[X](ω)=∑(n,k)∈INψn,k⋅Ξn,k(ω),
and we finally recognize in the previous expression that for all 0≤t≤1(4.55)ZtN=∑(n,k)∈INψn,k⋅Ξn,k=∑(n,k)∈INψn,k(t)⋅∫Uϕn,kT(s)dWs=∫01kN(t,s)dWs.We can now proceed to justify the main result of Proposition4.12.Proof.
The finite-dimensional processesZN defined through Lemma 4.13 have sample paths t↦ZtN(ω) belonging to ℰN. Moreover, for almost every ω in Ω  and for all n,k in ℐN,
(4.56)P〈ψn,k,ZN(ω)〉=P〈ψn,k,∫01kN(t,s)dWs(ω)〉=P〈ψn,k,∑(p,q)∈INψp,q(ω)⋅∫01ϕp,qT(s)dWs(ω)〉=∫01ϕn,kT(s)dWs(ω),
because of the orthonormality property of ψn,k with respect to 〈,〉. As the previous equalities hold  for every N>0, the applications x↦𝒫〈ψn,k,x〉 can naturally be extended on xΩ′ by continuity. Therefore, it makes sense to write, for all (n,k) in ℐN, 𝒫〈ψn,k,ZN〉=limN→∞𝒫〈ψn,k,ZN〉=def𝒫〈ψn,k,X〉 even if   X is defined into a larger sample space than ℰ. In other words, we have
(4.57)P〈ψn,k,X〉=∫01ϕn,kT(s)dWs=Ξn,k,
and we can thus express the conditioned process ZN=𝔼N[X] as the orthogonal projection of X onto the finite sample path ℰN by writing
(4.58)ZN=∑(n,k)∈INψn,k⋅P〈ψn,k,X〉.


## 4.2.3. Optimality Criterion of the Sample Paths

Proposition4.12 elucidates the structure of the conditioned processes ZN as pathwise orthogonal projections of X on the finite-dimensional RKHS ℰN. It allows us to cast the finite sample paths in a geometric setting and incidentally, to give a characterization of them as the minimizer of some functionals. In doing so, we shed a new light on well-known results of the interpolation theory [50–52] and extend them to the multidimensional case.The central point of this section reads as follows.Proposition 4.14.
Given a functionx in ℰ, the function xN=(Ψ∘ΔN)[x] belongs to ℰN and is defined by the following optimal criterion: xN is the only function in ℰ interpolating x on DN such that the functional
(4.59)〈y,y〉=‖L[y](t)‖22=∫01|L[y](t)|22dt
takes its unique minimal value over ℰ in xN.Proof.
The spaceℰN has been defined as ℰN=ΨN[ξΩN]=Ψ∘ΔN[ℰ] so that, for all x in ℰ, xN clearly belongs to ℰN. Moreover, xN interpolates x on DN: indeed, we know that the finite-dimensional operators ΔN and ΨN-1 are inverse of each other ΔN=ΨN-1, which entails that for all t in DN(4.60)xN(t)=(Ψ∘ΔN)[x](t)=(ΨN∘ΔN)[x](t)=x(t),
where we use the fact that, for any ξ in ξΩ′  and for all t in DN, ΨN[ξ](t)=Ψ[ξ](t) (recall that ψn,k(t)=0 if n>N and t belongs to DN).
Let us now show thatxN is determined in ℰ by the announced optimal criterion. Suppose y belongs to ℰ and interpolates x on DN, and remark that we can write
(4.61)〈y,y〉=‖L[y]‖22=‖(Φ∘Δ)[y](t)‖22=‖Δ[y]‖22,
since Φ is an isometry. Then, consider Δ[y] in l2(ξΩ) and remark that, since for all (n,k) in ℐN, δn,k are Dirac measures supported by DN, we have
(4.62)∀(n,k)∈IN,Δn,k[y]=P(δn,k,y)=P(δn,k,x)=Δn,k[x]=Δn,k[xN].
This entails
(4.63)‖Δ[y]‖22dt=∑(n,k)∈I|Δn,k[y]|22≥∑(n,k)∈IN|Δn,k[y]|22=‖Δ[xN]‖22dt.
Since, by definition of xN, δn,k[xN]=0 if n>N. Moreover, the minimum 〈xN,xN〉 is only attained for y such that δn,k[y]=0 if n>N and δn,k[y]=δn,k[x] if n≤N, which defines univocally xN. This establishes that, for all y in ℰ such that for all t in DN, y(t)=x(t), we have 〈xN,xN〉≤〈y,y〉 and the equality case holds if and only if y=xN.Remark 4.15.
Whenℒ represents a regular differential operator of order d, ∑i=1dai(t)Di, where D=d/dt, that is, for
(4.64)dXt=α(t)⋅Xt+Γ(t)⋅dWt,
with
(4.65)α(t)=[01⋱⋱⋱1adad-1…a1],Γ(t)=[00⋮1].
The finite-dimensional sample paths coincide exactly with the spline interpolation of order 2d+1, which are well known to satisfy the previous criterion [46]. This example will be further explored in the example section.The Dirichlet energy simply appears as the squared norm induced onℰ by the inner product 〈,〉, which in turn can be characterized as a Dirichlet quadratic form on ℰ. Actually, such a Dirichlet form can be used to define the Gauss-Markov process, extending the Gauss-Markov property to processes indexed on the multidimensional spaces parameter [19]. In particular, for an n-dimensional parameter space, we can condition such Gauss-Markov processes on a smooth n-1-dimensional boundary. Within the boundary, the sample paths of the resulting conditioned process (the solution to the prediction problem in [19]) are the solutions to the corresponding Dirichlet problems for the elliptic operator associated with the Dirichlet form.The characterization of the basis as the minimizer of such a Dirichlet energy (4.59) gives rise to an alternative method to compute the basis as the solution of a Dirichlet boundary value problem for an elliptic differential operator.Proposition 4.16.
Let us assume thatα and Γ are continuously differentiable and that Γ is invertible. Then, the functions μn,k are defined as
(4.66)μn,k(t)={μl(t),t∈[ln,k,mn,k],μr(t),t∈[mn,k,rn,k],0,else,
where μl and μr are the unique solutions of the second-order d-dimensional linear differential equation
(4.67)u′′+(Γ-1(αTΓ-Γ′)-α)u′-(Γ-1(αTΓ-Γ′)α+α′)u=0
with the following boundary value conditions:
(4.68)μl(ln,k)=0,μl(mn,k)=Id,μr(mn,k)=Id,μr(rn,k)=0.Proof.
By Proposition4.14, we know that μn,k(t) minimizes the convex functional
(4.69)∫01|L[u](s)|22ds
over ℰ, being equal to zero outside the interval [ln,k,rn,k] and equal to one at the point t=mn,k. Because of the hypotheses on α and Γ, we have ℒ=𝒟 and we can additionally restrain our search to functions that are twice continuously differentiable. Incidentally, we only need to minimize separately the contributions on the interval [ln,k,mn,k] and [mn,k,rn,k]. On both intervals, this problem is a classical Euler-Lagrange problem (see, e.g., [53]) and is solved using basic principles of calculus of variations. We easily identify the Lagrangian of our problem as
(4.70)L(t,u,u′)=|(u′-α(t)u(t))(Γ(t))-1|22=(u′(t)-α(t)u(t))T(Γ(t))-1(u′(t)-α(t)u(t)).
From there, after some simple matrix calculations, the Euler-Lagrange equations
(4.71)∂L(t,u,u′)∂ui-ddt(∂L(t,u,u′)∂u′i)=0,i=1,…,d,
can be expressed under the form:
(4.72)u′′+(Γ-1(αTΓ-Γ′)-α)u′-(Γ-1(αTΓ-Γ′)α+α′)u=0,
which ends the proof.Remark 4.17.
It is a simple matter of calculus to check that the expression ofμ given in Proposition 2.1 satisfies (4.67). Notice also that, in the case Γ=Id, the differential equation becomes
(4.73)u′′+(αT-α)u′-(αTα+α′)u=0,
which is further simplified for constant α.Under the hypotheses of Proposition4.16, we can thus define μn,k as the unique solution to the second-order linear differential equation (4.67) with the appropriate boundary values conditions. From this definition, it is then easy to derive the bases ψn,k by completing the following program.(1)
Compute thet↦μn,k(t) by solving the linear ordinary differential problem.(2)
Apply the differential operator𝒟 to get the functions 𝒟[μn,k].(3)
Orthonormalize the column functionst↦cj(𝒟[μn,k(t)]) by the Gram-Schmidt process.(4)
Apply the integral operator𝒦 to get the desired functions ψn,k (or equivalently multiply the original function t↦μn,k(t) by the corresponding Gram-Schmidt triangular matrix).Notice finally that each of these points is easily implemented numerically.


## 5. Examples: Derivation of the Bases for Some Classical Processes

### 5.1. One-Dimensional Case

In the one-dimensional case, the construction of the Gauss-Markov process is considerably simplified since we do not have to consider the potential degeneracy of matrix-valued functions. Indeed, in this situation, the centered Gauss-Markov processX is solution of the one-dimensional stochastic equation(5.1)dXt=α(t)Xtdt+Γ(t)dWt,
with α homogeneously Hölder continuous and Γ positive continuous function. We then have the Doob representation(5.2)Xt=g(t)∫0tf(s)dWs,withg(t)=e∫0tα(v)dv,f(t)=Γ(t)e-∫0tα(v)dv.
Writing the function h as(5.3)h(t)=∫0tf2(s)ds,
the covariance of the process reads for any 0≤t,s≤0(5.4)C(t,s)=g(t)g(s)h(t∧s).
The variance of the Gauss-Markov bridge Bt pinned in tx and tz yields(5.5)(σtx,tz(t))2=g(t)2(h(t)-h(tx))(h(tz)-h(t))h(tz)-h(tx).
These simple relations entail  that the functions ψn,k are defined on their supports Sn,k by ψn,k(t)2=𝔼[(δn(t))2] with(5.6)E[(δn(t))2]=(σln,k,rn,k(t))2-(1Sn+1,2k(t)(σln,k,mn,k(t))2+1Sn+1,2k+1(t)(σln,k,mn,k(t))2).This reads onSn+1,2k(5.7)ψn,k(t)2=g(t)2[(h(t)-h(ln,k))(h(rn,k)-h(t))h(rn,k)-h(ln,k)-(h(t)-h(ln,k))(h(mn,k)-h(t))h(mn,k)-h(ln,k)]
and on Sn+1,2k+1 as(5.8)ψn,k(t)2=g(t)2[(h(t)-h(ln,k))(h(rn,k)-h(t))h(rn,k)-h(ln,k)-(h(t)-h(mn,k))(h(rn,k)-h(t))h(rn,k)-h(mn,k)]
and therefore we have(5.9)ψn,k(t)={σn,kg(t)(h(t)-h(ln,k))g(mn,k)(h(mn,k)-h(ln,k)),ln,k≤t≤mn,k,σn,kg(t)(h(rn,k)-h(t))g(mn,k)(h(rn,k)-h(mn,k)),mn,k≤t≤rn,k,
with(5.10)σn,k=(h(rn,k)-h(mn,k))(h(mn,k)-h(ln,k))h(rn,k)-h(ln,k).
As for the first element, it simply results from the conditional expectation of the one-dimensional bridge pinned in l0,0=0 and r0,0=1:(5.11)ψ0,0(t)=g(t)(h(t)-h(l0,0))h(r0,0)-h(l0,0).
In this class of processes, two paradigmatic process  are the Wiener process and the Ornstein-Uhlenbeck processes with constant coefficients. In the case of the Wiener process, h(t)=t and g(t)=1, which yields the classical triangular-shaped   Schauder functions used by Lévy [8]. As for the Ornstein-Uhlenbeck process with constant coefficients α and Γ, we have g(t)=exp(αt), f(t)=Γexp(-αt) and h(t)=(Γ/2α)(1-e-2αt), yielding for the construction basis the expressions(5.12)ψn,k(t)={Γαsinh⁡(α(t-ln,k))sinh⁡(α(mn,k-ln,k)),ln,k≤t≤mn,k,Γαsinh⁡(α(rn,k-t))sinh⁡(α(mn,k-ln,k)),mn,k≤t≤rn,k,ψ0,0(t)=Γαe-α/2sinh⁡(αt)sinh⁡(α),
which were already evidenced in [54].


### 5.2. Multidimensional Case

In the multidimensional case, the explicit expressions for the basis functionsψn,k make fundamental use of the flow F of the underlying linear equation (2.3) for a given function α. For commutative forms of α (i.e., such that α(t)·α(s)=α(s)·α(t) for all t,s), the flow can be formally expressed as an exponential operator. It is, however,  a notoriously difficult problem to find a tractable expression for general α. As a consequence, it is only possible to provide closed-from formulae for our basis functions in very specific cases.


#### 5.2.1. Multidimensional Gauss-Markov Rotations

We consider in this section thatα is antisymmetric and constant and Γ∈ℝd×m such that Γ=σ2Id. For α antisymmetric, since αT(t)=-α(t), we have(5.13)F(s,t)T=F(s,t)-1,
that is, the flow is unitary. This property implies that(5.14)hu(s,t)=σ2∫stF(w,u)F(w,u)Tdw=σ2(t-s)Id,
which yields by definition of σn,k(5.15)σn,k⋅σn,kT=σ2(mn,k-ln,k)(rn,k-mn,k)rn,k-ln,kId.
The square root σn,k is then uniquely defined (by choosing both Cholesky and symmetrical square roots) by(5.16)σn,k=σ(mn,k-ln,k)(rn,k-mn,k)(rn,k-ln,k)Id,
and ψn,k(t) reads(5.17)ψn,k(t)={σrn,k-mn,k(mn,k-ln,k)(rn,k-ln,k)(t-ln,k)F(mn,k,t),ln,k≤t≤mn,k,σmn,k-ln,k(rn,k-mn,k)(rn,k-ln,k)(rn,k-t)F(mn,k,t),ln,k≤t≤mn,k.Recognizing the(n,k) element of the Schauder basis for the construction of the one-dimensional Wiener process(5.18)sn,k(t)={rn,k-mn,k(rn,k-ln,k)(mn,k-ln,k)(t-ln,k),ln,k≤t≤mn,k,mn,k-ln,k(rn,k-ln,k)(rn,k-mn,k)(rn,k-t),ln,k≤t≤mn,k,
we obtain the following formula:(5.19)ψn,k(t)=σsn,k(t)F(t-mn,k).This form shows that the Schauder basis for multidimensional rotations results from the multiplication of the triangular-shaped elementary function used for the Lévy-Ciesielski construction of the Wiener process with the flow of the equation, that is, the elementary rotation.The simplest example of this kind is the stochastic sine and cosine process corresponding to(5.20)α=(01-10),Γ=σ2I2.
In that case, ψn,k has the expression(5.21)ψn,k(t)=sn,k(t)(cos⁡(t-mn,k)-sin⁡(t-mn,k)sin⁡(t-mn,k)cos⁡(t-mn,k)).
Interestingly, the different basis functions have the structure of the solutions of the nonstochastic oscillator equation. One of the equations perturbs the trajectory in the radial component of the deterministic solution  and the other one in the tangential direction. We represent such a construction in Figure 2 with the additional conditioning that X1=X0, that is, imposing that the trajectory forms a loop between time 0 and 1.Figure 2
Construction of the stochastic sine and cos Ornstein-Uhelenbeck processes for the parameters given in (5.20): multiresolution construction of the sample path.(a) Basis for the construction of the Integrated Wiener process (d=2). Plain red curve: ψ1,1, dashed red: ψ2,1, plain blue: ψ1,2 and dashed blue: ψ2,2. (b) 10-steps construction of the process. Moreover, we observe that each basis accounts separately for different aspects of sample path: ψ2,1 fixes the value of the integrated process at the middle point mn,k and ψ2,2 the value of the derivative of the process at the endpoints {ln,k,rn,k} in relationship with the value of the Wiener process at the middle point, whose contributions are split between functions ψ1,1 and ψ1,2 (see Figure 3).
(a)(b)


#### 5.2.2. The Successive Primitives of the Wiener Process

In applications, it often occurs that people use smooth stochastic processes to model the integration of noisy signals. This is for instance the case of a particular subject of a Brownian forcing or of the synaptic integration of noisy inputs [55]. Such smooth processes involves in general integrated martingales, and the simplest example of such processes are the successive primitives of a standard Wiener process.Letd>2, and denote by Xtd the d-1th order primitive of the Wiener process. This process can be defined via the lower-order primitives Xtk for k<d via the relations(5.22)dXtk+1=Xtkdt,k<d,dXt1=dWt,
where Wt is a standard real Wiener process. These equations can be written in our formalism as(5.23)dXt=α(t)⋅Xt+Γ(t)⋅dWt,
with(5.24)α(t)=[01⋱⋱⋱10],Γ(t)=[00⋮1].
In particular, though none of the integrated processes Xk for K>1 is Markov by itself, the d-uplet X=(Xd,…,X1) is a Gauss-Markov process.Furthermore, because of the simplicity and the sparsity of the matrices involved, we can identify in a compact form all the variables used in the computation of the construction basis for these processes. In particular, the flowF of the equation is the exponential of the matrix α, and since α is nilpotent, it is easy to show that F has the expression,(5.25)F(s,t)=[1(t-s)(t-s)22⋯(t-s)d-1(d-1)!⋱⋱⋱⋮⋱⋱(t-s)22⋱(t-s)1]
and the only nonzero entry of the d×d matrix Γ is one at position (d-1,d-1). Using this expression and the highly simple expression of Γ, we can compute the general element of the matrix hu(t,s), which reads(5.26)(hu(s,t))i,j=(-1)i+j(t-u)2d-1-(i+j)-(s-u)2d-1-(i+j)(2d-1-(i+j))(d-1-i)!(d-1-j)!.
Eventually, we observe that the functions ψn,k, yielding the multiresolution description of the integrated Wiener processes, are directly deduced from the matrix-valued function(5.27)(cn,k(t))i,j={ψn,k⋅Ln,k-1=g(t)h(ln,k,t),ln,k≤t≤mn,k,ψn,k⋅Rn,k-1=g(t)h(t,rn,k),mn,k≤t≤rn,k,
whose components are further expressed as(5.28)(cn,k(t))i,j=∑p=id-1(-1)p+jti-p(i-p)!t2d-1-(p+j)-ln,k2d-1-(p+j)(2d-1-(p+j))(d-1-p)!(d-1-j)!,
for ln,k≤t≤mn,k and as(5.29)(cn,k(t))i,j=∑p=id-1(-1)p+jti-p(i-p)!mn,k2d-1-(p+j)-t2d-1-(p+j)(2d-1-(p+j))(d-1-p)!(d-1-j)!,
for mn,k≤t≤rn,k. The final computation of the ψn,k involves the computation of Ln,k and Rn,k, which in the general case can become very complex. However, this expression is highly simplified when assuming that mn,k is the middle of the interval [ln,k,rn,k]. Indeed, in that case, we observe that, for any (i,j) such that i+j is odd, (hm(l,r))i,j=0, which induces the same property on the covariance matrix Σn,k and on the polynomials (cn,k(t))i,j. This property gives therefore a preference to the dyadic partition that provides simple expressions for the basis elements in any dimensions, and allows simple computations of the basis.Remark 5.1.
Observe that, for all0≤i<d-1, we have
(5.30)(cn,k(t))i,j′=(cn,k(t))i+1,j±∑p=id-1(-1)p+jti-p(i-p)!t2d-1-(p+j)-ln,k2d-2-(p+j)(d-1-p)!(d-1-j)!,=(cn,k(t))i+1,j±td-j-1(d-j-1)!∑q=0d-1-i(-t)qt(d-1-i)-pp!((d-1-i)-p)!︸0.
As Ln,k and Rn,k are constant, we immediately deduce the important relation that, for all 0≤i≤d-1,(ψn,k(t))0,j(i)=(ψn,k(t))i,j. This indicates that each finite-dimensional sample paths of our construction has components that satisfies the nondeterministic equation associated with the iteratively integrated Wiener process. Actually, this fact is better stated remembering that the Schauder basis ψn,k and the corresponding orthonormal basis ϕn,k:[0,1]→ℝ1×d are linked through (3.10), which reads
(5.31)[(ψn,k)0,0′…(ψn,k)0,d-1′⋮⋮(ψn,k)d-2,0′…(ψn,k)d-2,d-1′(ψn,k)d-1,0′…(ψn,k)d-1,d-1′]=[(ψn,k)1,0…(ψn,k)1,d-1⋮⋮(ψn,k)d-1,0…(ψn,k)d-1,d-10…0]+[0…0⋮⋮0…0(ψn,k)0,0…(ψn,k)0,d-1].
Additionally, we realize that the orthonormal basis is entirely determined by the one-dimensional families (ϕn,k)0,j, which are mutually orthogonal functions satisfying (ϕn,k)0,j=(ψn,k)0,j(d).We study in more detail the case of the integrated and doubly-integrated Wiener process (d=2 and d=3), for which closed-form expressions are provided in Appendices A and B. As expected, the first row of the basis functions for the integrated Wiener process turns out to be the well-known cubic Hermite splines [56]. These functions have been widely used in numerical analysis and actually constitute the basis of the lowest degree in a wider family of bases known as the natural basis of polynomial splines of interpolation [25]. Such bases are used to interpolate data points with constraint of smoothness of different degrees (e.g., the cubic Hermite splines ensure that the resulting interpolation is in C1[0,1]). The next family of splines of interpolation (corresponding to the C2 constraint) is naturally retrieved by considering the construction of the doubly-integrated Wiener process: we obtain a family of three 3-dimensional functions  that constitutes the columns of a 3×3 matrix that we denote by ψ. The top row is made of polynomials of degree five, which have again simple expressions when mn,k is the middle of the interval [ln,k,rn,k].


## 5.1. One-Dimensional Case

In the one-dimensional case, the construction of the Gauss-Markov process is considerably simplified since we do not have to consider the potential degeneracy of matrix-valued functions. Indeed, in this situation, the centered Gauss-Markov processX is solution of the one-dimensional stochastic equation(5.1)dXt=α(t)Xtdt+Γ(t)dWt,
with α homogeneously Hölder continuous and Γ positive continuous function. We then have the Doob representation(5.2)Xt=g(t)∫0tf(s)dWs,withg(t)=e∫0tα(v)dv,f(t)=Γ(t)e-∫0tα(v)dv.
Writing the function h as(5.3)h(t)=∫0tf2(s)ds,
the covariance of the process reads for any 0≤t,s≤0(5.4)C(t,s)=g(t)g(s)h(t∧s).
The variance of the Gauss-Markov bridge Bt pinned in tx and tz yields(5.5)(σtx,tz(t))2=g(t)2(h(t)-h(tx))(h(tz)-h(t))h(tz)-h(tx).
These simple relations entail  that the functions ψn,k are defined on their supports Sn,k by ψn,k(t)2=𝔼[(δn(t))2] with(5.6)E[(δn(t))2]=(σln,k,rn,k(t))2-(1Sn+1,2k(t)(σln,k,mn,k(t))2+1Sn+1,2k+1(t)(σln,k,mn,k(t))2).This reads onSn+1,2k(5.7)ψn,k(t)2=g(t)2[(h(t)-h(ln,k))(h(rn,k)-h(t))h(rn,k)-h(ln,k)-(h(t)-h(ln,k))(h(mn,k)-h(t))h(mn,k)-h(ln,k)]
and on Sn+1,2k+1 as(5.8)ψn,k(t)2=g(t)2[(h(t)-h(ln,k))(h(rn,k)-h(t))h(rn,k)-h(ln,k)-(h(t)-h(mn,k))(h(rn,k)-h(t))h(rn,k)-h(mn,k)]
and therefore we have(5.9)ψn,k(t)={σn,kg(t)(h(t)-h(ln,k))g(mn,k)(h(mn,k)-h(ln,k)),ln,k≤t≤mn,k,σn,kg(t)(h(rn,k)-h(t))g(mn,k)(h(rn,k)-h(mn,k)),mn,k≤t≤rn,k,
with(5.10)σn,k=(h(rn,k)-h(mn,k))(h(mn,k)-h(ln,k))h(rn,k)-h(ln,k).
As for the first element, it simply results from the conditional expectation of the one-dimensional bridge pinned in l0,0=0 and r0,0=1:(5.11)ψ0,0(t)=g(t)(h(t)-h(l0,0))h(r0,0)-h(l0,0).
In this class of processes, two paradigmatic process  are the Wiener process and the Ornstein-Uhlenbeck processes with constant coefficients. In the case of the Wiener process, h(t)=t and g(t)=1, which yields the classical triangular-shaped   Schauder functions used by Lévy [8]. As for the Ornstein-Uhlenbeck process with constant coefficients α and Γ, we have g(t)=exp(αt), f(t)=Γexp(-αt) and h(t)=(Γ/2α)(1-e-2αt), yielding for the construction basis the expressions(5.12)ψn,k(t)={Γαsinh⁡(α(t-ln,k))sinh⁡(α(mn,k-ln,k)),ln,k≤t≤mn,k,Γαsinh⁡(α(rn,k-t))sinh⁡(α(mn,k-ln,k)),mn,k≤t≤rn,k,ψ0,0(t)=Γαe-α/2sinh⁡(αt)sinh⁡(α),
which were already evidenced in [54].


## 5.2. Multidimensional Case

In the multidimensional case, the explicit expressions for the basis functionsψn,k make fundamental use of the flow F of the underlying linear equation (2.3) for a given function α. For commutative forms of α (i.e., such that α(t)·α(s)=α(s)·α(t) for all t,s), the flow can be formally expressed as an exponential operator. It is, however,  a notoriously difficult problem to find a tractable expression for general α. As a consequence, it is only possible to provide closed-from formulae for our basis functions in very specific cases.


### 5.2.1. Multidimensional Gauss-Markov Rotations

We consider in this section thatα is antisymmetric and constant and Γ∈ℝd×m such that Γ=σ2Id. For α antisymmetric, since αT(t)=-α(t), we have(5.13)F(s,t)T=F(s,t)-1,
that is, the flow is unitary. This property implies that(5.14)hu(s,t)=σ2∫stF(w,u)F(w,u)Tdw=σ2(t-s)Id,
which yields by definition of σn,k(5.15)σn,k⋅σn,kT=σ2(mn,k-ln,k)(rn,k-mn,k)rn,k-ln,kId.
The square root σn,k is then uniquely defined (by choosing both Cholesky and symmetrical square roots) by(5.16)σn,k=σ(mn,k-ln,k)(rn,k-mn,k)(rn,k-ln,k)Id,
and ψn,k(t) reads(5.17)ψn,k(t)={σrn,k-mn,k(mn,k-ln,k)(rn,k-ln,k)(t-ln,k)F(mn,k,t),ln,k≤t≤mn,k,σmn,k-ln,k(rn,k-mn,k)(rn,k-ln,k)(rn,k-t)F(mn,k,t),ln,k≤t≤mn,k.Recognizing the(n,k) element of the Schauder basis for the construction of the one-dimensional Wiener process(5.18)sn,k(t)={rn,k-mn,k(rn,k-ln,k)(mn,k-ln,k)(t-ln,k),ln,k≤t≤mn,k,mn,k-ln,k(rn,k-ln,k)(rn,k-mn,k)(rn,k-t),ln,k≤t≤mn,k,
we obtain the following formula:(5.19)ψn,k(t)=σsn,k(t)F(t-mn,k).This form shows that the Schauder basis for multidimensional rotations results from the multiplication of the triangular-shaped elementary function used for the Lévy-Ciesielski construction of the Wiener process with the flow of the equation, that is, the elementary rotation.The simplest example of this kind is the stochastic sine and cosine process corresponding to(5.20)α=(01-10),Γ=σ2I2.
In that case, ψn,k has the expression(5.21)ψn,k(t)=sn,k(t)(cos⁡(t-mn,k)-sin⁡(t-mn,k)sin⁡(t-mn,k)cos⁡(t-mn,k)).
Interestingly, the different basis functions have the structure of the solutions of the nonstochastic oscillator equation. One of the equations perturbs the trajectory in the radial component of the deterministic solution  and the other one in the tangential direction. We represent such a construction in Figure 2 with the additional conditioning that X1=X0, that is, imposing that the trajectory forms a loop between time 0 and 1.Figure 2
Construction of the stochastic sine and cos Ornstein-Uhelenbeck processes for the parameters given in (5.20): multiresolution construction of the sample path.(a) Basis for the construction of the Integrated Wiener process (d=2). Plain red curve: ψ1,1, dashed red: ψ2,1, plain blue: ψ1,2 and dashed blue: ψ2,2. (b) 10-steps construction of the process. Moreover, we observe that each basis accounts separately for different aspects of sample path: ψ2,1 fixes the value of the integrated process at the middle point mn,k and ψ2,2 the value of the derivative of the process at the endpoints {ln,k,rn,k} in relationship with the value of the Wiener process at the middle point, whose contributions are split between functions ψ1,1 and ψ1,2 (see Figure 3).
(a)(b)


### 5.2.2. The Successive Primitives of the Wiener Process

In applications, it often occurs that people use smooth stochastic processes to model the integration of noisy signals. This is for instance the case of a particular subject of a Brownian forcing or of the synaptic integration of noisy inputs [55]. Such smooth processes involves in general integrated martingales, and the simplest example of such processes are the successive primitives of a standard Wiener process.Letd>2, and denote by Xtd the d-1th order primitive of the Wiener process. This process can be defined via the lower-order primitives Xtk for k<d via the relations(5.22)dXtk+1=Xtkdt,k<d,dXt1=dWt,
where Wt is a standard real Wiener process. These equations can be written in our formalism as(5.23)dXt=α(t)⋅Xt+Γ(t)⋅dWt,
with(5.24)α(t)=[01⋱⋱⋱10],Γ(t)=[00⋮1].
In particular, though none of the integrated processes Xk for K>1 is Markov by itself, the d-uplet X=(Xd,…,X1) is a Gauss-Markov process.Furthermore, because of the simplicity and the sparsity of the matrices involved, we can identify in a compact form all the variables used in the computation of the construction basis for these processes. In particular, the flowF of the equation is the exponential of the matrix α, and since α is nilpotent, it is easy to show that F has the expression,(5.25)F(s,t)=[1(t-s)(t-s)22⋯(t-s)d-1(d-1)!⋱⋱⋱⋮⋱⋱(t-s)22⋱(t-s)1]
and the only nonzero entry of the d×d matrix Γ is one at position (d-1,d-1). Using this expression and the highly simple expression of Γ, we can compute the general element of the matrix hu(t,s), which reads(5.26)(hu(s,t))i,j=(-1)i+j(t-u)2d-1-(i+j)-(s-u)2d-1-(i+j)(2d-1-(i+j))(d-1-i)!(d-1-j)!.
Eventually, we observe that the functions ψn,k, yielding the multiresolution description of the integrated Wiener processes, are directly deduced from the matrix-valued function(5.27)(cn,k(t))i,j={ψn,k⋅Ln,k-1=g(t)h(ln,k,t),ln,k≤t≤mn,k,ψn,k⋅Rn,k-1=g(t)h(t,rn,k),mn,k≤t≤rn,k,
whose components are further expressed as(5.28)(cn,k(t))i,j=∑p=id-1(-1)p+jti-p(i-p)!t2d-1-(p+j)-ln,k2d-1-(p+j)(2d-1-(p+j))(d-1-p)!(d-1-j)!,
for ln,k≤t≤mn,k and as(5.29)(cn,k(t))i,j=∑p=id-1(-1)p+jti-p(i-p)!mn,k2d-1-(p+j)-t2d-1-(p+j)(2d-1-(p+j))(d-1-p)!(d-1-j)!,
for mn,k≤t≤rn,k. The final computation of the ψn,k involves the computation of Ln,k and Rn,k, which in the general case can become very complex. However, this expression is highly simplified when assuming that mn,k is the middle of the interval [ln,k,rn,k]. Indeed, in that case, we observe that, for any (i,j) such that i+j is odd, (hm(l,r))i,j=0, which induces the same property on the covariance matrix Σn,k and on the polynomials (cn,k(t))i,j. This property gives therefore a preference to the dyadic partition that provides simple expressions for the basis elements in any dimensions, and allows simple computations of the basis.Remark 5.1.
Observe that, for all0≤i<d-1, we have
(5.30)(cn,k(t))i,j′=(cn,k(t))i+1,j±∑p=id-1(-1)p+jti-p(i-p)!t2d-1-(p+j)-ln,k2d-2-(p+j)(d-1-p)!(d-1-j)!,=(cn,k(t))i+1,j±td-j-1(d-j-1)!∑q=0d-1-i(-t)qt(d-1-i)-pp!((d-1-i)-p)!︸0.
As Ln,k and Rn,k are constant, we immediately deduce the important relation that, for all 0≤i≤d-1,(ψn,k(t))0,j(i)=(ψn,k(t))i,j. This indicates that each finite-dimensional sample paths of our construction has components that satisfies the nondeterministic equation associated with the iteratively integrated Wiener process. Actually, this fact is better stated remembering that the Schauder basis ψn,k and the corresponding orthonormal basis ϕn,k:[0,1]→ℝ1×d are linked through (3.10), which reads
(5.31)[(ψn,k)0,0′…(ψn,k)0,d-1′⋮⋮(ψn,k)d-2,0′…(ψn,k)d-2,d-1′(ψn,k)d-1,0′…(ψn,k)d-1,d-1′]=[(ψn,k)1,0…(ψn,k)1,d-1⋮⋮(ψn,k)d-1,0…(ψn,k)d-1,d-10…0]+[0…0⋮⋮0…0(ψn,k)0,0…(ψn,k)0,d-1].
Additionally, we realize that the orthonormal basis is entirely determined by the one-dimensional families (ϕn,k)0,j, which are mutually orthogonal functions satisfying (ϕn,k)0,j=(ψn,k)0,j(d).We study in more detail the case of the integrated and doubly-integrated Wiener process (d=2 and d=3), for which closed-form expressions are provided in Appendices A and B. As expected, the first row of the basis functions for the integrated Wiener process turns out to be the well-known cubic Hermite splines [56]. These functions have been widely used in numerical analysis and actually constitute the basis of the lowest degree in a wider family of bases known as the natural basis of polynomial splines of interpolation [25]. Such bases are used to interpolate data points with constraint of smoothness of different degrees (e.g., the cubic Hermite splines ensure that the resulting interpolation is in C1[0,1]). The next family of splines of interpolation (corresponding to the C2 constraint) is naturally retrieved by considering the construction of the doubly-integrated Wiener process: we obtain a family of three 3-dimensional functions  that constitutes the columns of a 3×3 matrix that we denote by ψ. The top row is made of polynomials of degree five, which have again simple expressions when mn,k is the middle of the interval [ln,k,rn,k].


## 5.2.1. Multidimensional Gauss-Markov Rotations

We consider in this section thatα is antisymmetric and constant and Γ∈ℝd×m such that Γ=σ2Id. For α antisymmetric, since αT(t)=-α(t), we have(5.13)F(s,t)T=F(s,t)-1,
that is, the flow is unitary. This property implies that(5.14)hu(s,t)=σ2∫stF(w,u)F(w,u)Tdw=σ2(t-s)Id,
which yields by definition of σn,k(5.15)σn,k⋅σn,kT=σ2(mn,k-ln,k)(rn,k-mn,k)rn,k-ln,kId.
The square root σn,k is then uniquely defined (by choosing both Cholesky and symmetrical square roots) by(5.16)σn,k=σ(mn,k-ln,k)(rn,k-mn,k)(rn,k-ln,k)Id,
and ψn,k(t) reads(5.17)ψn,k(t)={σrn,k-mn,k(mn,k-ln,k)(rn,k-ln,k)(t-ln,k)F(mn,k,t),ln,k≤t≤mn,k,σmn,k-ln,k(rn,k-mn,k)(rn,k-ln,k)(rn,k-t)F(mn,k,t),ln,k≤t≤mn,k.Recognizing the(n,k) element of the Schauder basis for the construction of the one-dimensional Wiener process(5.18)sn,k(t)={rn,k-mn,k(rn,k-ln,k)(mn,k-ln,k)(t-ln,k),ln,k≤t≤mn,k,mn,k-ln,k(rn,k-ln,k)(rn,k-mn,k)(rn,k-t),ln,k≤t≤mn,k,
we obtain the following formula:(5.19)ψn,k(t)=σsn,k(t)F(t-mn,k).This form shows that the Schauder basis for multidimensional rotations results from the multiplication of the triangular-shaped elementary function used for the Lévy-Ciesielski construction of the Wiener process with the flow of the equation, that is, the elementary rotation.The simplest example of this kind is the stochastic sine and cosine process corresponding to(5.20)α=(01-10),Γ=σ2I2.
In that case, ψn,k has the expression(5.21)ψn,k(t)=sn,k(t)(cos⁡(t-mn,k)-sin⁡(t-mn,k)sin⁡(t-mn,k)cos⁡(t-mn,k)).
Interestingly, the different basis functions have the structure of the solutions of the nonstochastic oscillator equation. One of the equations perturbs the trajectory in the radial component of the deterministic solution  and the other one in the tangential direction. We represent such a construction in Figure 2 with the additional conditioning that X1=X0, that is, imposing that the trajectory forms a loop between time 0 and 1.Figure 2
Construction of the stochastic sine and cos Ornstein-Uhelenbeck processes for the parameters given in (5.20): multiresolution construction of the sample path.(a) Basis for the construction of the Integrated Wiener process (d=2). Plain red curve: ψ1,1, dashed red: ψ2,1, plain blue: ψ1,2 and dashed blue: ψ2,2. (b) 10-steps construction of the process. Moreover, we observe that each basis accounts separately for different aspects of sample path: ψ2,1 fixes the value of the integrated process at the middle point mn,k and ψ2,2 the value of the derivative of the process at the endpoints {ln,k,rn,k} in relationship with the value of the Wiener process at the middle point, whose contributions are split between functions ψ1,1 and ψ1,2 (see Figure 3).
(a)(b)


## 5.2.2. The Successive Primitives of the Wiener Process

In applications, it often occurs that people use smooth stochastic processes to model the integration of noisy signals. This is for instance the case of a particular subject of a Brownian forcing or of the synaptic integration of noisy inputs [55]. Such smooth processes involves in general integrated martingales, and the simplest example of such processes are the successive primitives of a standard Wiener process.Letd>2, and denote by Xtd the d-1th order primitive of the Wiener process. This process can be defined via the lower-order primitives Xtk for k<d via the relations(5.22)dXtk+1=Xtkdt,k<d,dXt1=dWt,
where Wt is a standard real Wiener process. These equations can be written in our formalism as(5.23)dXt=α(t)⋅Xt+Γ(t)⋅dWt,
with(5.24)α(t)=[01⋱⋱⋱10],Γ(t)=[00⋮1].
In particular, though none of the integrated processes Xk for K>1 is Markov by itself, the d-uplet X=(Xd,…,X1) is a Gauss-Markov process.Furthermore, because of the simplicity and the sparsity of the matrices involved, we can identify in a compact form all the variables used in the computation of the construction basis for these processes. In particular, the flowF of the equation is the exponential of the matrix α, and since α is nilpotent, it is easy to show that F has the expression,(5.25)F(s,t)=[1(t-s)(t-s)22⋯(t-s)d-1(d-1)!⋱⋱⋱⋮⋱⋱(t-s)22⋱(t-s)1]
and the only nonzero entry of the d×d matrix Γ is one at position (d-1,d-1). Using this expression and the highly simple expression of Γ, we can compute the general element of the matrix hu(t,s), which reads(5.26)(hu(s,t))i,j=(-1)i+j(t-u)2d-1-(i+j)-(s-u)2d-1-(i+j)(2d-1-(i+j))(d-1-i)!(d-1-j)!.
Eventually, we observe that the functions ψn,k, yielding the multiresolution description of the integrated Wiener processes, are directly deduced from the matrix-valued function(5.27)(cn,k(t))i,j={ψn,k⋅Ln,k-1=g(t)h(ln,k,t),ln,k≤t≤mn,k,ψn,k⋅Rn,k-1=g(t)h(t,rn,k),mn,k≤t≤rn,k,
whose components are further expressed as(5.28)(cn,k(t))i,j=∑p=id-1(-1)p+jti-p(i-p)!t2d-1-(p+j)-ln,k2d-1-(p+j)(2d-1-(p+j))(d-1-p)!(d-1-j)!,
for ln,k≤t≤mn,k and as(5.29)(cn,k(t))i,j=∑p=id-1(-1)p+jti-p(i-p)!mn,k2d-1-(p+j)-t2d-1-(p+j)(2d-1-(p+j))(d-1-p)!(d-1-j)!,
for mn,k≤t≤rn,k. The final computation of the ψn,k involves the computation of Ln,k and Rn,k, which in the general case can become very complex. However, this expression is highly simplified when assuming that mn,k is the middle of the interval [ln,k,rn,k]. Indeed, in that case, we observe that, for any (i,j) such that i+j is odd, (hm(l,r))i,j=0, which induces the same property on the covariance matrix Σn,k and on the polynomials (cn,k(t))i,j. This property gives therefore a preference to the dyadic partition that provides simple expressions for the basis elements in any dimensions, and allows simple computations of the basis.Remark 5.1.
Observe that, for all0≤i<d-1, we have
(5.30)(cn,k(t))i,j′=(cn,k(t))i+1,j±∑p=id-1(-1)p+jti-p(i-p)!t2d-1-(p+j)-ln,k2d-2-(p+j)(d-1-p)!(d-1-j)!,=(cn,k(t))i+1,j±td-j-1(d-j-1)!∑q=0d-1-i(-t)qt(d-1-i)-pp!((d-1-i)-p)!︸0.
As Ln,k and Rn,k are constant, we immediately deduce the important relation that, for all 0≤i≤d-1,(ψn,k(t))0,j(i)=(ψn,k(t))i,j. This indicates that each finite-dimensional sample paths of our construction has components that satisfies the nondeterministic equation associated with the iteratively integrated Wiener process. Actually, this fact is better stated remembering that the Schauder basis ψn,k and the corresponding orthonormal basis ϕn,k:[0,1]→ℝ1×d are linked through (3.10), which reads
(5.31)[(ψn,k)0,0′…(ψn,k)0,d-1′⋮⋮(ψn,k)d-2,0′…(ψn,k)d-2,d-1′(ψn,k)d-1,0′…(ψn,k)d-1,d-1′]=[(ψn,k)1,0…(ψn,k)1,d-1⋮⋮(ψn,k)d-1,0…(ψn,k)d-1,d-10…0]+[0…0⋮⋮0…0(ψn,k)0,0…(ψn,k)0,d-1].
Additionally, we realize that the orthonormal basis is entirely determined by the one-dimensional families (ϕn,k)0,j, which are mutually orthogonal functions satisfying (ϕn,k)0,j=(ψn,k)0,j(d).We study in more detail the case of the integrated and doubly-integrated Wiener process (d=2 and d=3), for which closed-form expressions are provided in Appendices A and B. As expected, the first row of the basis functions for the integrated Wiener process turns out to be the well-known cubic Hermite splines [56]. These functions have been widely used in numerical analysis and actually constitute the basis of the lowest degree in a wider family of bases known as the natural basis of polynomial splines of interpolation [25]. Such bases are used to interpolate data points with constraint of smoothness of different degrees (e.g., the cubic Hermite splines ensure that the resulting interpolation is in C1[0,1]). The next family of splines of interpolation (corresponding to the C2 constraint) is naturally retrieved by considering the construction of the doubly-integrated Wiener process: we obtain a family of three 3-dimensional functions  that constitutes the columns of a 3×3 matrix that we denote by ψ. The top row is made of polynomials of degree five, which have again simple expressions when mn,k is the middle of the interval [ln,k,rn,k].


## 6. Stochastic Calculus from the Hilbert Point of View

Thus far, all calculations, propositions, and theorems are valid for any finite-dimensional  the Gauss-Markov process and all the results are valid pathwise, that is, for each sample path. The analysis provides a Hilbert description of the processes as a series of standard Gaussian random variables multiplied by certain specific functions, that form a Schauder basis in the suitable spaces. This new description of Gauss-Markov processes provides a new way for treating problems arising in the study of stochastic processes. As examples of this, we derive the Itô formula and the Girsanov theorem from the Hilbertian viewpoint. Note that these results are equalities in law, that is, dealing with the distribution of stochastic processes, which is a weaker notion compared to the pathwise analysis. In this section, we restrict our analysis to the one-dimensional case for technical simplicity.The closed-form expressions of the basis of functionsψn,k in the one-dimensional case are given in Section 5.1. The differential and integral operators associated, introduced in Section 3.1.2 are highly simplified in the one-dimensional case. Let U be a bounded open set of [0,1], we denote by C(U) the space of continuous real functions on U and we recall that the topological dual of C(U) is R(U), the space of Radon measures on U. We also introduce D0(U), the space of test functions in C∞(U) that are zero in zero, and whose dual space D0′(U) satisfies D0′(U)⊂R(U).Let U be a bounded open neighborhood of [0,1], and denote by C(U) is the space of continuous real functions on U, R(U) its topological dual, the space of Radon measures, D0(U) the space of test function in C∞(U) which are zero at zero and it dual D0′(U)⊂R(U). We consider the Gelfand triple(6.1)D0(U)⊂C(U)⊂L2(U)⊂D0′(U)⊂R(U).The integral operator𝒦 is defined (and extended by dual pairing) by(6.2)K[⋅](t)=∫U1[0,t](s)gα(t)fα(s)⋅ds,
and the inverse differential operator 𝒟 reads(6.3)D[⋅](t)=1gα(t)ddt(⋅fα(t)).Now that we dispose of all the explicit forms of the basis functions and related operators, we are in position to complete our program  and start by proving the very important Itô formula and its finite-dimensional counterpart before turning to the Girsanov  theorem.


### 6.1. Itô’s Formula

A very useful theorem in the stochastic processes theory is the Itô formula. We show here that this formula is consistent with the Hilbert framework introduced. Most of the proofs can be found in AppendixE. The proof of Itô formula is based on demonstrating the integration by parts property.Proposition 6.1 (Integration by parts).
Let(Xt) and (Yt) be two one-dimensional Gauss-Markov processes starting from zero. Then one has the following equality in law:
(6.4)XtYt=∫0tXs∘dYs+∫0tYs∘dXs,
where, for  two stochastic processes At and Bt, ∫0tAs∘dBs denotes the Stratonovich integral. In terms of the Itô integral, this formula is written as
(6.5)XtYt=∫0tXsdYs+∫0tYsdXs+〈X,Y〉t,
where the brackets denote the mean quadratic variation.The proof of this proposition is quite technical and is provided in AppendixE. It is based a thorough analysis of the finite-dimensional processes XtN and YtN. For this integration by parts formula and using a density argument, one can recover the more general Itô formula.Theorem 6.2 (Itô).
Let(Xt)t be a Gauss-Markov process and F∈C2(ℝ). The process f(Xt) is a Markov process and satisfies the following relation in law:
(6.6)f(Xt)=f(X0)+∫0tf′(Xs)dXs+12∫0tf′′(Xs)d〈X〉s.This theorem is proved in AppendixE.The Itô  formula implies in particular that the multiresolution description developed in the paper is valid for every smooth functional of a Gauss-Markov process. In particular, it allows a simple description of exponential functionals of Gaussian Markovian processes, which are of particular interest in mathematics and have many applications, in particular in economics (see, e.g., [57]).Therefore, we observe that in the view of the paper, Itô  formula stems from the nonorthogonal projections of basis element. For multidimensional processes, the proof of the Itô formula is deduced from the one-dimensional proof and would involve the study of the multidimensional bridge formula forXt and Yt.We eventually remark that this section provides us with a finite-dimensional counterpart of the Itô  formula for discretized processes, which has important potential applications, and further assesses the suitability of using the finite resolution representation developed in this paper. Indeed, using the framework developed in the present paper allows considering finite-resolution processes and their transformation through nonlinear smooth transformation in a way that is concordant with the standard stochastic calculus processes, since the equation on the transformed process indeed converges towards its Itô representation as the resolution increases.


### 6.2. Girsanov Formula: A Geometric Viewpoint

In the framework we developed, transforming a processX into a process Y is equivalent to substituting the Schauder construction basis related to Y for the basis related to X. Such an operation provides a pathwise mapping for each sample path of X onto a sample path of Y having the same probability density in ξΩ′. This fact sheds a new light on the geometry of multidimensional Gauss-Markov processes, since the relationship between two processes is seen as a linear change of basis. In our framework, this relationship between processes is straightforwardly studied in the finite rank approximations of the processes up to a certain resolution. Technical intricacy is nevertheless raised when dealing with the representation of the process itself in the infinite-dimensional Hilbert spaces. We solve these technical issues here  and show that in the limit N→∞ one recovers Girsanov  theorem as a limit of the linear transformations between the Gauss-Markov processes.The general problem consists therefore in studying the relationship between two real Gauss-Markov processesX and Y that are defined by(6.7)dXt=αX(t)Xtdt+ΓX(t)dWt,dYt=αY(t)Xtdt+ΓY(t)dWt.We have noticed that the spacesxΩ′ are the same in the one-dimensional case as long as both ΓX and ΓY never vanish  and therefore make this assumption here. In order to further simplify the problem, we assume that γX,Y=ΓX/ΓY is continuously differentiable. This assumption allows us to introduce the process Zt=γX,Y(t)Yt that satisfies the stochastic differential equation(6.8)dZt=ddt(γX,Y(t))Ytdt+γX,Y(t)dYt=(ddt(γX,Y(t))+γX,Y(t)αY(t))Ytdt+ΓX(t)dWt=αZ(t)Ztdt+ΓX(t)dWt,
with αZ(t)=(d/dt)(γX,Y(t))γX,Y(t)-1+αY(t). Moreover, if Zψn,k and Yψn,k are the bases of functions that describe the process Z and Y, respectively, we have Zψn,k=γX,Y·Yψn,k.The previous remarks allow us to restrict without loss of generality our study to  the processes defined for  same functionΓ, thus reducing the parameterization of the  Gauss-Markov processes to the linear coefficient α. Observe that in the classical stochastic calculus theory, it is well known that such hypotheses are necessary for the process X to be absolutely continuous with respect to Y (through the use of the Girsanov  theorem).Let us now consider thatα,β, and Γ three real Hölder continuous real functions, and introduce αX and βX solutions of the equations(6.9)d(αXt)=α(t)(αXt)dt+Γ(t)dWt,d(βXt)=β(t)(βXt)dt+Γ(t)dWt.
All the functions and tools related to the process αX (resp., βX) will be indexed by α (β)  in the sequel.


#### 6.2.1. Lift Operators

Depending on the space we  are considering (either coefficients or trajectories), we define the two following operators mapping the processαXt on βXt.(1)
Thecoefficients lift operator α,βG is the linear operator mapping in ξΩ′ the process αX on the process βX:
(6.10)α,βG=Δβ∘Ψα:(ξΩ′,B(ξΩ′))⟶(ξΩ′,B(ξΩ′)).
For any ξ∈ξΩ′, the operator α,βG maps a sample path of αX on a sample path of βX.(2)
Theprocess lift operator α,βF is the linear operator mapping in xΩ the process αX on the process βX:
(6.11)α,βH=αΨ∘αΔ:(xΩ,B(xΩ))⟶(xΩ,B(xΩ)).
We summarize the properties of these operators now.Proposition 6.3.
The operatorsα,βG and α,βH satisfy the following properties.(i)
They are linear measurable bijections(ii)
For everyN>0, the function α,βGN=PN∘α,βG∘IN:ξΩN′→ξΩN (resp., α,βHN=PN∘α,βH∘IN:ξΩN′→ξΩN) is a finite-dimensional linear operator, whose matrix representation is triangular in the natural basis of ξΩN (resp., ξΩN′) and whose eigenvalues α,βνn,k are given by
(6.12)α,βνn,k=gα(mn,k)gβ(mn,k)βMn,kαMn,k,0≤n≤N,0≤k<2N-1
(resp., β,ανn,k=(α,βνn,k)-1).(iii)
α,βG and α,βH are bounded operators for the spectral norm with
(6.13)‖α,βG‖2=sup⁡nsup⁡kνα,βn,k≤sup⁡gαinf⁡gβsup⁡fα2inf⁡fβ2<∞,
and ∥α,βH∥2=∥β,αG∥2<∞.(iv)
The determinants ofα,βGN (denoted by α,βJN) and α,βHN admit a limit when N tends to infinity:
(6.14)Jα,β=lim⁡N→∞Jα,βN=exp⁡(12(∫01(α(t)-β(t))dt)),lim⁡N→∞det⁡(Hα,βN)=exp⁡(12(∫01(β(t)-α(t))dt))=Jβ,α.The proof of these properties elementary stems from the analysis done on the functionsΨ and Δ that were previously performed, and these are detailed in Appendix C.


#### 6.2.2. Radon-Nikodym Derivatives

From the properties proved on the lift operators, we are in position to further analyze the relationship between the probability distributions ofαX and βX. We first consider the finite-dimensional processes αXN and βXN. We emphasize that,  throughout this section, all equalities are true pathwise.Lemma 6.4.
Given the finite-dimensional measuresPαN and PβN, the Radon-Nikodym derivative of PβN with respect to PαN satisfies
(6.15)dPβNdPαN(ω)=Jα,βN⋅exp⁡(-12(ΞN(ω)T(Sα,βN-IdξΩN)ΞN(ω)))
with α,βSN=α,βGNT·α,βGN and the equality is true pathwise.Proof.
In the finite-dimensional case, for allN>0, we can write that PαN, PβNpαN, and the Lebesgue measure xΩN are mutually absolutely continuous: we denote by pαN and pβN the Gaussian density of PαN and PβN with respect to the Lebesgue measure on xΩN. Therefore, in the finite-dimensional case, the Radon-Nikodym derivative of PβN with respect to PαN is defined  to be pathwise and is simply given by the quotient of the density of the vector {βXN(mi,j)} with the density of the vector {αXN(mi,j)} for 0≤i≤N, 0≤j<2i-1, that is,
(6.16)dPβNdPαN(ω)=pβN(αXN(ω))pαN(αXN(ω))=det⁡(αΣN)det⁡(βΣN)⋅exp⁡(-12(αXN(ω)T(βΣN-1-αΣN-1)αXN(ω))).
We first make explicit
(6.17)det⁡(αΣN)det⁡(βΣN)=det⁡(αΣN⋅βΣN-1)=det⁡(αΨN⋅αΨNT⋅βΔNT⋅βΔN)=det⁡(βΔN⋅αΨN⋅αΨNT⋅βΔNT)=det⁡(α,βGN⋅α,βGNT)=det⁡(α,βGN)2.
Then, we rearrange the exponent using the Cholesky decomposition:
(6.18)ΣβN-1-ΣαN-1=ΔβNT⋅ΔβN-ΔαNT⋅ΔαN,
so that we write the exponent of (6.16) as
(6.19)αXNT(ΔβNT⋅ΔβN-ΔαNT⋅ΔαN)XαN=ΞNT⋅ΨαNT(ΔβNT⋅ΔβN-αΔNT⋅ΔαN)ΨαN(ω)⋅ΞN=ΞNT(Gα,βNT⋅Gα,βN-IdξΩN)ΞN.
We finally reformulate (6.16) as
(6.20)dPβNdPαN(ω)=Jα,βN⋅exp⁡(-12(ΞN(ω)T(α,βGNT⋅α,βGN-IdξΩN)ΞN(ω))).Let us now justify from a geometrical point of view why this formula is a direct consequence of the finite-dimensional change of variable formula on the model spaceξΩN. If we introduce αΔN, the coefficient application related to αXN, we know that ΞN=αΔN(αXN) follows a normal law 𝒩(0,IξΩN). We denote by pξN its standard Gaussian density with respect to the Lebesgue measure on ξΩN. We also know that(6.21)αΔN(βXN)=(αΔN∘α,βFN)(αXN)=β,αGN(αΔN(αXN))=β,αGN(ΞN).Sinceβ,αGN is linear, the change of the variable formula directly entails that β,αGN(ΞN) admits on ξΩN(6.22)pβ,αN(ξN)=|det⁡(α,βG)|pξN[ξNT(α,βGNT⋅α,βGN)ξN]
as density with respect to the Lebesgue measure. Consider now B as a measurable set of (xΩN,ℬ(xΩN)); then we have(6.23)PβN(B)=∫αΔN(B)pβ,αN(ξN)dξN=∫αΔN(B)pβ,αN(ξN)pξN(ξN)pξN(ξN)dξN=∫Bpβ,αN(αΔN(XN))pξN(αΔN(XN))dPαN(XN),
from which we immediately conclude.


#### 6.2.3. The Trace Class Operator

The pathwise expression of the Radon-Nikodym derivative extends to the infinite-dimensional representation ofαX and βX. This extension involves technical analysis on the infinite-dimensional Hilbert space l2(ℝ). We have shown in Proposition 6.3 that the application α,βG was bounded for the spectral norm. Therefore, we have, for any ξ in l2(ℝ), the inequality(6.24)‖Gα,β(ξ)‖2≤‖Gα,β‖2⋅‖ξ‖2
implying that α,βG maps l2(ℝ) into l2(ℝ). We can then define the adjoint operator α,βGT from l2(ℝ) to l2(ℝ), which is given by(6.25)∀ξ,η∈l2(R),(Gα,βT(η),ξ)=(η,Gα,β(ξ)).
Let us now consider the self-adjoint operator α,βS=Gα,βT∘Gα,β:l2(ℝ)→l2(ℝ). This operator is the infinite-dimensional counterpart of the matrix α,βSN.Lemma 6.5.
Considering the coefficients of the matrix representation ofSα,β in the natural basis en,k of l2(ℝ) given as α,βSp,qn,k=(en,k,Sα,β(ep,q)), one has
(6.26)Sα,βp,qn,k=∫01(ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t))(ϕαp,q(t)+(α(t)-β(t))Γ(t)ψαp,q(t))dt.Proof.
Assume thatmax(n,p)≤N and that (n,k) and (p,q)∈ℐN. With the notations used previously with U, an open neighbourhood of [0,1], we have
(6.27)α,βSp,qn,k=(α,βG(en,k),α,βG(ep,q))=∑(i,j)∈I∫Uδβi,j(t)ψαn,k(t)dt∫Uδβi,j(t)ψαp,q(t)dt.
Since we have by definition αψn,k=𝒦α[αϕn,k], δβi,j=𝒟β[ϕβi,j],   𝒦α-1=𝒟α*,   𝒟β-1=𝒦β* on the space D0′(U), we have
(6.28)βϕi,j(t)=Kβ*[δβi,j](t)=fβ(t)∫Ugβ(s)δβi,j(s)ds,(6.29)αϕn,k(t)=Dα*[αψn,k](t)=1fα(t)ddt(αψn,k(t)gα(t)).
From there using (6.28), we can write the integration by part formula in the sense of the generalized functions to get
(6.30)∫Uδβi,j(t)αψn,k(t)dt=∫U1fβ(t)ddt(αψn,k(t)gβ(t))ϕβi,j(t)dt=∫UDβ*[ψαn,k](t)ϕβi,j(t)dt.
We now compute using (6.29):
(6.31)Dβ*[ψαn,k](t)=1fβ(t)ddt(ψαn,k(t)gα(t)gα(t)gβ(t))=gα(t)f(t)gβ(t)fβ(t)ϕα(t)-ddt(gα(t)gβ(t))ψαn,k(t)gβ(t)fβ(t).
Specifying gα, gβ, fα, and fβ and recalling the relations
(6.32)gα(t)fα(t)=gβ(t)fβ(t)=Γ(t),ddt(gα(t)gβ(t))=(α(t)-β(t))gα(t)gβ(t),
we rewrite the function (6.31) in L2[0,1] as
(6.33)βD*[ψαn,k](t)=1fβ(t)ddt(αψn,k(t)gβ(t))=ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t).
Now, since the family βϕn,k forms a complete orthonormal system of L2[0,1] and is zero outside [0,1], by the Parseval identity, expression (6.27) can be written as the scalar product:
(6.34)α,βSp,qn,k=∫01(ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t))(ϕαp,q(t)+(α(t)-β(t))Γ(t)ψαp,q(t))dt,
which ends the proof of the lemma.Notice that we can further simplify expression (6.26):(6.35)α,βSp,qn,k=δp,qn,k+∫01(α(t)-β(t))2Γ(t)(ψαn,k(t)ψαp,q(t))dt+∫01(α(t)-β(t))Γ(t)(ϕαn,k(t)ψαp,q(t)+ϕαp,q(t)ψαn,k(t))dt.
We are now in a position to show that the operator α,βS-Id can be seen as the limit of the finite-dimensional operator α,βSN-IdξΩN, in the following sense.Theorem 6.6.
The operatorα,βS-Id:l2(ℝ)→l2(ℝ) is a trace class operator, whose trace is given by
(6.36)Tr⁡(Sα,β-Id)=∫01(α(t)-β(t))dt+∫01hα(t)fα(t)2(α(t)-β(t))2dt.We prove this essential point in AppendixF. The proof consists in showing that the operator α,βS-Id is isometric to a Hilbert-Schmidt operator whose trace can be computed straightforwardly.


#### 6.2.4. The Girsanov Theorem

We now proceed to prove the Girsanov theorem by extending the domain of the quadratic form associated withα,βS-Id to the space ξΩ′, which can only be done in law.Theorem 6.7.
In the infinite-dimensional case, the Radon-Nikodym derivative ofPβ=PβX-1 with respect to Pα=PαX-1 reads
(6.37)dPβ(ω)dPα(ω)=exp⁡(-12(∫01β(t)-α(t)dt+Ξ(ω)T(α,βS-IdξΩ′)Ξ(ω))),
which in terms of the Itô stochastic integral reads
(6.38)dPβ(ω)dPα(ω)=exp⁡⁡(∫01β(t)-α(t)fα2(t)αXt(ω)gα(t)d(αXt(ω)gα(t)))-12∫01(β(t)-α(t))2fα2(t)(αXt(ω)gα(t))2dt.In order to demonstrate the Girsanov theorem from our geometrical point of view, we need to establish the following result.Lemma 6.8.
The positive definite quadratic form onl2(ℝ)×l2(ℝ) associated with operator α,βS-Id:l2(ℝ)→l2(ℝ) is well defined on ξΩ′. Moreover, for all ξΩ′,
(6.39)(ξ,(α,βS-IdξΩ′)(ξ))=2∫01α(t)-β(t)fα2(t)αXt(ξ)gα(t)∘d(αXt(ξ)gα(t))+∫01(α(t)-β(t))2fα2(t)(αXt(ξ)gα(t))2dt,
where αXt(ξ)=αΦ(ξ) and ∘ refers to the Stratonovich integral and the equality is true in law.Proof of Theorem6.7.
We start by writing the finite-dimensional Radon-Nikodym derivative(6.40)dPβ(ω)dPα(ω)=JNα,β⋅lim⁡N→∞exp⁡(-12(ΞN(ω)T(α,βSN-IdξΩN)ΞN(ω))).
By Proposition 6.3, we have
(6.41)Jα,β=lim⁡N→∞Jα,βN=12∫01(α(t)-β(t))dt.
If, as usual, Ξ denotes a recursively indexed infinite-dimensional vector  of independent variables with law 𝒩(0,1) and ΞN=ξPN∘Ξ, writing ξn,k=Ξn,k(ω), we have
(6.42)ΞN(ω)T(α,βSN-IdξΩN)ΞN(ω)=∑n=0N∑p=0N∑0≤k<2n-1∑0≤q<2p-1ξn,k[α,βSN-IdξΩN]p,qn,kξp,q.
We know that ξ is almost surely in ξΩ′, and, by Lemma 6.8, we also know that on ξΩ′×Ωξ′(6.43)lim⁡N→∞(ξ,(Sα,βN-IdξΩN)(ξ))=(ξ,(Sα,β-IdΩξ′)(ξ)),
so that we can effectively write the infinite-dimensional Radon-Nikodym derivative on ξΩ′ as the point-wise limit of the finite-dimensional one  on ξΩN through the projectors ξPN:
(6.44)dPβdPα(ω)=lim⁡N→∞dPβNdPαN(ω),
which directly yields formula (6.37).
The derivation of the Girsanov formula (6.40) from (6.37) comes from the relationship between the Stratonovich and Itô formulas  since the quadratic variation of αXt/gα(t) and (α(t)-β(t))/fα2(t)×αXt/gα(t) reads
(6.45)∫01E(∫0tfα(s)dWs,α(t)-β(t)fα2(t)∫0tfα(s)dWs)=∫01(α(t)-β(t))dt.
Therefore, the expression of the Radon-Nikodym derivative in Lemma 6.8 can be written in terms of the Itô integrals as
(6.46)(ξ,(α,βS-IdξΩ′)(ξ))=∫01(α(t)-β(t))dt+2∫01α(t)-β(t)f2(t)αXt(ξ)gα(t)d(αXt(ξ)gα(t))+∫01(α(t)-β(t))2f2(t)(αXt(ξ)gα(t))2dt
and the Radon-Nikodym derivative as
(6.47)dPβdPα(ω)=exp⁡⁡(∫01β(t)-α(t)fα2(t)αXt(ω)gα(t)d(αXt(ω)gα(t))-12∫01(β(t)-α(t))2fα2(t)(αXt(ω)gα(t))2dt).Observe that, ifα(t)=0, we recover the familiar expression(6.48)dPβdPα(ω)=exp⁡(∫01β(t)Γ(t)Wt(ω)dWt(ω)-12∫01β(t)2Γ(t)W(ω)t2dt).Conclusion and Perspectives
The discrete construction we present displays both analytical and numerical interests for further applications. From the analysis viewpoint, even if the basis does not exhibit the same orthogonal properties as the Karhunen-Loève decomposition, it has the important advantage of saving the structure of sample paths through its property of strong pathwise convergence and of providing a multiscale representation of the processes, which contrasts with the convergence in the mean of the Karhunen-Loève decomposition. From the numerical viewpoint, three Haar-like properties make our decomposition particularly suitable for certain numerical computations: (i) all basis elements have compact support on an open interval that has the structure of dyadic rational endpoints, (ii) these intervals are nested and become smaller for larger indices of the basis element, and (iii) for any interval endpoint, only a finite number of basis elements are nonzero at that point. Thus the expansion in our basis, when evaluated at an interval endpoint (e.g., dyadic rational), terminates in a finite number of steps. Moreover, the very nature of the construction based on an increasingly refined description of the sample paths paves the way to coarse-graining approaches similar to wavelet decompositions in signal processing. In view of this, our framework offers promising applications.Dichotomic Search of First-Hitting Times
The first application we envisage concerns the problem of first-hitting times. Because of its manifold applications, finding the time when a process first exits a given region  is a central question of stochastic calculus. However, closed-form theoretical results are scarce and one often has to resort to numerical algorithms [59]. In this regard, the multiresolution property suggests an exact scheme to simulate sample paths of a Gaussian Markov process X in an iterative “top-down’’ fashion. Assuming the intervals are dyadic rational  and that we have a conditional knowledge of a sample path on the dyadic points of DN={k2-N∣0≤k≤2N}, one can decide to further the simulation of this sample path at any time t in DN+1 by drawing a point according to the conditional law of Xt given {Xt}t∈DN, which is simply expressed in the framework of our construction. This property can be used for great advantages in numerical computations such as dichotomic search algorithms for first passage times: the key element is to find an estimate of the true conditional probability that a hitting time has occurred when knowing the value of the process at two given times, one in the past and one in the future. With such an estimate, an efficient strategy to look for passage times consists in refining the sample path when and only when its trajectory is estimated likely to actually cross the barrier. Thus the sample path of the process is represented at poor temporal resolution when it is far from the boundary and at increasingly higher resolution closer to the boundary. Such an algorithmic principle achieves a high level of precision in the computation of the first-hitting time, while demanding far less operation than usual stochastic Runge-Kutta scheme. This approach has been successfully implemented for the one-dimensional case [58], see Figure 4. In that article, the precision of the algorithm is controlled as well as the probability to evaluate a first hitting time substantially different from the actual value. The approach proves to be extremely efficient compared to customary methods. The general multidimensional approach proposed in the present paper allows direct generalization of these results to the computation of exit times in any dimension and for general smooth sets [30–32].Gaussian Deformation Modes in Nonlinear Diffusions
The present study is developed for the Gauss-Markov systems. However, many models arising in applied science present nonlinearities, and in that case, the construction based on a sum of Gaussian random variables will not generalize. However, the Gaussian case treated here can nevertheless be applied to perturbation of nonlinear differential equations with small noise. LetF:ℝ×ℝd↦ℝd be a nonlinear time-varying vector field, and let us assume that X0(t) is a stable (attractive) solution of the dynamical system:
(6.49)dXdt=F(t,X).
This function X0 can for instance be a fixed point (in which case it is a constant), a cycle (in which case it is periodic), or a general attractive orbit of the system. In the deterministic case, any solution having its initial condition in a given neighbourhood ℬ in ℝ×ℝd of the solution will asymptotically converge towards the solution, and therefore perturbations of the solutions are bounded. Let us now consider that the system is subject to a small amount of noise and define Y∈ℝd as the solution of the stochastic nonlinear differential equation:
(6.50)dYt=F(t,Yt)dt+ɛΓ(t,Yt)dWt.
Assuming that the noise is small (i.e., ɛ is a small parameter), because of the attractivity of the solution X0(t), the function Y(t) will remain very close to X0(t) (at least in a bounded time interval). In this region, we define ɛZt=Yt-X0(t). This stochastic variable is the solution of the equation
(6.51)dZt=1ɛ(F(t,X0(t)+ɛZt)-F(t,X0(t))+ɛΓ(t,X0(t)+ɛZt)dWt)=(∇xF)(t,X0(t))Zt+Γ(t,X0(t))dWt+O(ɛ).
The solution at the first order in ɛ is therefore the multidimensional Gaussian process with nonconstant coefficients:
(6.52)dZt=(∇xF)(t,X0(t))Zt+Γ(t,X0(t))dWt,
and our theory describes the solutions in a multiresolution framework. Notice that, in that perspective, our basis functions ψn,k can be seen as increasingly finer modes of deformation of a deterministic trajectory. This approach appears particularly relevant to the theory of weakly interconnected neural oscillators in computational neuroscience [60]. Indeed, one of the most popular approaches of this field, the phase model theory, formally consists in studying how perturbations are integrated in the neighborhood of an attracting cycle [61, 62].
All these instances are exemplary of how our multiresolution description of the Gauss-Markov processes offers a simple yet rigorous tool to broach a large number of open problems and promises fascinating applications both in theoretical and in applied science.Figure 4
First passage time for an Ornstein-Uhlenbeck processU with a elastic coefficient α=1 and a noise intensity Γ=1, the barrier is constant Λ=1 and the initial condition is U0=0. The plots represent at different scales a realization Ut(ω) for which the algorithm of [58] returns a first passage time τ(ω)=8.00469684 with a resolution δt=221=5107 after 5 recursive calls. The whole sample path is represented in (a) and a series of zooms is carried out around τ(ω) in (b), (c), and (d). The dilation coefficients are set according to the scale invariance of a Wiener process with the time scale being expanded by 20 and the distance scale by 45 during each dilation. The series (b), (c), and (c), (d) zoom on regions where the sample path gets close to the barrier. The simulation of the sample path has required 683 subdivisions and illustrates an unfavorable situation since the expected number of divisions is approximatively 284 for this particular setting.


## 6.1. Itô’s Formula

A very useful theorem in the stochastic processes theory is the Itô formula. We show here that this formula is consistent with the Hilbert framework introduced. Most of the proofs can be found in AppendixE. The proof of Itô formula is based on demonstrating the integration by parts property.Proposition 6.1 (Integration by parts).
Let(Xt) and (Yt) be two one-dimensional Gauss-Markov processes starting from zero. Then one has the following equality in law:
(6.4)XtYt=∫0tXs∘dYs+∫0tYs∘dXs,
where, for  two stochastic processes At and Bt, ∫0tAs∘dBs denotes the Stratonovich integral. In terms of the Itô integral, this formula is written as
(6.5)XtYt=∫0tXsdYs+∫0tYsdXs+〈X,Y〉t,
where the brackets denote the mean quadratic variation.The proof of this proposition is quite technical and is provided in AppendixE. It is based a thorough analysis of the finite-dimensional processes XtN and YtN. For this integration by parts formula and using a density argument, one can recover the more general Itô formula.Theorem 6.2 (Itô).
Let(Xt)t be a Gauss-Markov process and F∈C2(ℝ). The process f(Xt) is a Markov process and satisfies the following relation in law:
(6.6)f(Xt)=f(X0)+∫0tf′(Xs)dXs+12∫0tf′′(Xs)d〈X〉s.This theorem is proved in AppendixE.The Itô  formula implies in particular that the multiresolution description developed in the paper is valid for every smooth functional of a Gauss-Markov process. In particular, it allows a simple description of exponential functionals of Gaussian Markovian processes, which are of particular interest in mathematics and have many applications, in particular in economics (see, e.g., [57]).Therefore, we observe that in the view of the paper, Itô  formula stems from the nonorthogonal projections of basis element. For multidimensional processes, the proof of the Itô formula is deduced from the one-dimensional proof and would involve the study of the multidimensional bridge formula forXt and Yt.We eventually remark that this section provides us with a finite-dimensional counterpart of the Itô  formula for discretized processes, which has important potential applications, and further assesses the suitability of using the finite resolution representation developed in this paper. Indeed, using the framework developed in the present paper allows considering finite-resolution processes and their transformation through nonlinear smooth transformation in a way that is concordant with the standard stochastic calculus processes, since the equation on the transformed process indeed converges towards its Itô representation as the resolution increases.


## 6.2. Girsanov Formula: A Geometric Viewpoint

In the framework we developed, transforming a processX into a process Y is equivalent to substituting the Schauder construction basis related to Y for the basis related to X. Such an operation provides a pathwise mapping for each sample path of X onto a sample path of Y having the same probability density in ξΩ′. This fact sheds a new light on the geometry of multidimensional Gauss-Markov processes, since the relationship between two processes is seen as a linear change of basis. In our framework, this relationship between processes is straightforwardly studied in the finite rank approximations of the processes up to a certain resolution. Technical intricacy is nevertheless raised when dealing with the representation of the process itself in the infinite-dimensional Hilbert spaces. We solve these technical issues here  and show that in the limit N→∞ one recovers Girsanov  theorem as a limit of the linear transformations between the Gauss-Markov processes.The general problem consists therefore in studying the relationship between two real Gauss-Markov processesX and Y that are defined by(6.7)dXt=αX(t)Xtdt+ΓX(t)dWt,dYt=αY(t)Xtdt+ΓY(t)dWt.We have noticed that the spacesxΩ′ are the same in the one-dimensional case as long as both ΓX and ΓY never vanish  and therefore make this assumption here. In order to further simplify the problem, we assume that γX,Y=ΓX/ΓY is continuously differentiable. This assumption allows us to introduce the process Zt=γX,Y(t)Yt that satisfies the stochastic differential equation(6.8)dZt=ddt(γX,Y(t))Ytdt+γX,Y(t)dYt=(ddt(γX,Y(t))+γX,Y(t)αY(t))Ytdt+ΓX(t)dWt=αZ(t)Ztdt+ΓX(t)dWt,
with αZ(t)=(d/dt)(γX,Y(t))γX,Y(t)-1+αY(t). Moreover, if Zψn,k and Yψn,k are the bases of functions that describe the process Z and Y, respectively, we have Zψn,k=γX,Y·Yψn,k.The previous remarks allow us to restrict without loss of generality our study to  the processes defined for  same functionΓ, thus reducing the parameterization of the  Gauss-Markov processes to the linear coefficient α. Observe that in the classical stochastic calculus theory, it is well known that such hypotheses are necessary for the process X to be absolutely continuous with respect to Y (through the use of the Girsanov  theorem).Let us now consider thatα,β, and Γ three real Hölder continuous real functions, and introduce αX and βX solutions of the equations(6.9)d(αXt)=α(t)(αXt)dt+Γ(t)dWt,d(βXt)=β(t)(βXt)dt+Γ(t)dWt.
All the functions and tools related to the process αX (resp., βX) will be indexed by α (β)  in the sequel.


### 6.2.1. Lift Operators

Depending on the space we  are considering (either coefficients or trajectories), we define the two following operators mapping the processαXt on βXt.(1)
Thecoefficients lift operator α,βG is the linear operator mapping in ξΩ′ the process αX on the process βX:
(6.10)α,βG=Δβ∘Ψα:(ξΩ′,B(ξΩ′))⟶(ξΩ′,B(ξΩ′)).
For any ξ∈ξΩ′, the operator α,βG maps a sample path of αX on a sample path of βX.(2)
Theprocess lift operator α,βF is the linear operator mapping in xΩ the process αX on the process βX:
(6.11)α,βH=αΨ∘αΔ:(xΩ,B(xΩ))⟶(xΩ,B(xΩ)).
We summarize the properties of these operators now.Proposition 6.3.
The operatorsα,βG and α,βH satisfy the following properties.(i)
They are linear measurable bijections(ii)
For everyN>0, the function α,βGN=PN∘α,βG∘IN:ξΩN′→ξΩN (resp., α,βHN=PN∘α,βH∘IN:ξΩN′→ξΩN) is a finite-dimensional linear operator, whose matrix representation is triangular in the natural basis of ξΩN (resp., ξΩN′) and whose eigenvalues α,βνn,k are given by
(6.12)α,βνn,k=gα(mn,k)gβ(mn,k)βMn,kαMn,k,0≤n≤N,0≤k<2N-1
(resp., β,ανn,k=(α,βνn,k)-1).(iii)
α,βG and α,βH are bounded operators for the spectral norm with
(6.13)‖α,βG‖2=sup⁡nsup⁡kνα,βn,k≤sup⁡gαinf⁡gβsup⁡fα2inf⁡fβ2<∞,
and ∥α,βH∥2=∥β,αG∥2<∞.(iv)
The determinants ofα,βGN (denoted by α,βJN) and α,βHN admit a limit when N tends to infinity:
(6.14)Jα,β=lim⁡N→∞Jα,βN=exp⁡(12(∫01(α(t)-β(t))dt)),lim⁡N→∞det⁡(Hα,βN)=exp⁡(12(∫01(β(t)-α(t))dt))=Jβ,α.The proof of these properties elementary stems from the analysis done on the functionsΨ and Δ that were previously performed, and these are detailed in Appendix C.


### 6.2.2. Radon-Nikodym Derivatives

From the properties proved on the lift operators, we are in position to further analyze the relationship between the probability distributions ofαX and βX. We first consider the finite-dimensional processes αXN and βXN. We emphasize that,  throughout this section, all equalities are true pathwise.Lemma 6.4.
Given the finite-dimensional measuresPαN and PβN, the Radon-Nikodym derivative of PβN with respect to PαN satisfies
(6.15)dPβNdPαN(ω)=Jα,βN⋅exp⁡(-12(ΞN(ω)T(Sα,βN-IdξΩN)ΞN(ω)))
with α,βSN=α,βGNT·α,βGN and the equality is true pathwise.Proof.
In the finite-dimensional case, for allN>0, we can write that PαN, PβNpαN, and the Lebesgue measure xΩN are mutually absolutely continuous: we denote by pαN and pβN the Gaussian density of PαN and PβN with respect to the Lebesgue measure on xΩN. Therefore, in the finite-dimensional case, the Radon-Nikodym derivative of PβN with respect to PαN is defined  to be pathwise and is simply given by the quotient of the density of the vector {βXN(mi,j)} with the density of the vector {αXN(mi,j)} for 0≤i≤N, 0≤j<2i-1, that is,
(6.16)dPβNdPαN(ω)=pβN(αXN(ω))pαN(αXN(ω))=det⁡(αΣN)det⁡(βΣN)⋅exp⁡(-12(αXN(ω)T(βΣN-1-αΣN-1)αXN(ω))).
We first make explicit
(6.17)det⁡(αΣN)det⁡(βΣN)=det⁡(αΣN⋅βΣN-1)=det⁡(αΨN⋅αΨNT⋅βΔNT⋅βΔN)=det⁡(βΔN⋅αΨN⋅αΨNT⋅βΔNT)=det⁡(α,βGN⋅α,βGNT)=det⁡(α,βGN)2.
Then, we rearrange the exponent using the Cholesky decomposition:
(6.18)ΣβN-1-ΣαN-1=ΔβNT⋅ΔβN-ΔαNT⋅ΔαN,
so that we write the exponent of (6.16) as
(6.19)αXNT(ΔβNT⋅ΔβN-ΔαNT⋅ΔαN)XαN=ΞNT⋅ΨαNT(ΔβNT⋅ΔβN-αΔNT⋅ΔαN)ΨαN(ω)⋅ΞN=ΞNT(Gα,βNT⋅Gα,βN-IdξΩN)ΞN.
We finally reformulate (6.16) as
(6.20)dPβNdPαN(ω)=Jα,βN⋅exp⁡(-12(ΞN(ω)T(α,βGNT⋅α,βGN-IdξΩN)ΞN(ω))).Let us now justify from a geometrical point of view why this formula is a direct consequence of the finite-dimensional change of variable formula on the model spaceξΩN. If we introduce αΔN, the coefficient application related to αXN, we know that ΞN=αΔN(αXN) follows a normal law 𝒩(0,IξΩN). We denote by pξN its standard Gaussian density with respect to the Lebesgue measure on ξΩN. We also know that(6.21)αΔN(βXN)=(αΔN∘α,βFN)(αXN)=β,αGN(αΔN(αXN))=β,αGN(ΞN).Sinceβ,αGN is linear, the change of the variable formula directly entails that β,αGN(ΞN) admits on ξΩN(6.22)pβ,αN(ξN)=|det⁡(α,βG)|pξN[ξNT(α,βGNT⋅α,βGN)ξN]
as density with respect to the Lebesgue measure. Consider now B as a measurable set of (xΩN,ℬ(xΩN)); then we have(6.23)PβN(B)=∫αΔN(B)pβ,αN(ξN)dξN=∫αΔN(B)pβ,αN(ξN)pξN(ξN)pξN(ξN)dξN=∫Bpβ,αN(αΔN(XN))pξN(αΔN(XN))dPαN(XN),
from which we immediately conclude.


### 6.2.3. The Trace Class Operator

The pathwise expression of the Radon-Nikodym derivative extends to the infinite-dimensional representation ofαX and βX. This extension involves technical analysis on the infinite-dimensional Hilbert space l2(ℝ). We have shown in Proposition 6.3 that the application α,βG was bounded for the spectral norm. Therefore, we have, for any ξ in l2(ℝ), the inequality(6.24)‖Gα,β(ξ)‖2≤‖Gα,β‖2⋅‖ξ‖2
implying that α,βG maps l2(ℝ) into l2(ℝ). We can then define the adjoint operator α,βGT from l2(ℝ) to l2(ℝ), which is given by(6.25)∀ξ,η∈l2(R),(Gα,βT(η),ξ)=(η,Gα,β(ξ)).
Let us now consider the self-adjoint operator α,βS=Gα,βT∘Gα,β:l2(ℝ)→l2(ℝ). This operator is the infinite-dimensional counterpart of the matrix α,βSN.Lemma 6.5.
Considering the coefficients of the matrix representation ofSα,β in the natural basis en,k of l2(ℝ) given as α,βSp,qn,k=(en,k,Sα,β(ep,q)), one has
(6.26)Sα,βp,qn,k=∫01(ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t))(ϕαp,q(t)+(α(t)-β(t))Γ(t)ψαp,q(t))dt.Proof.
Assume thatmax(n,p)≤N and that (n,k) and (p,q)∈ℐN. With the notations used previously with U, an open neighbourhood of [0,1], we have
(6.27)α,βSp,qn,k=(α,βG(en,k),α,βG(ep,q))=∑(i,j)∈I∫Uδβi,j(t)ψαn,k(t)dt∫Uδβi,j(t)ψαp,q(t)dt.
Since we have by definition αψn,k=𝒦α[αϕn,k], δβi,j=𝒟β[ϕβi,j],   𝒦α-1=𝒟α*,   𝒟β-1=𝒦β* on the space D0′(U), we have
(6.28)βϕi,j(t)=Kβ*[δβi,j](t)=fβ(t)∫Ugβ(s)δβi,j(s)ds,(6.29)αϕn,k(t)=Dα*[αψn,k](t)=1fα(t)ddt(αψn,k(t)gα(t)).
From there using (6.28), we can write the integration by part formula in the sense of the generalized functions to get
(6.30)∫Uδβi,j(t)αψn,k(t)dt=∫U1fβ(t)ddt(αψn,k(t)gβ(t))ϕβi,j(t)dt=∫UDβ*[ψαn,k](t)ϕβi,j(t)dt.
We now compute using (6.29):
(6.31)Dβ*[ψαn,k](t)=1fβ(t)ddt(ψαn,k(t)gα(t)gα(t)gβ(t))=gα(t)f(t)gβ(t)fβ(t)ϕα(t)-ddt(gα(t)gβ(t))ψαn,k(t)gβ(t)fβ(t).
Specifying gα, gβ, fα, and fβ and recalling the relations
(6.32)gα(t)fα(t)=gβ(t)fβ(t)=Γ(t),ddt(gα(t)gβ(t))=(α(t)-β(t))gα(t)gβ(t),
we rewrite the function (6.31) in L2[0,1] as
(6.33)βD*[ψαn,k](t)=1fβ(t)ddt(αψn,k(t)gβ(t))=ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t).
Now, since the family βϕn,k forms a complete orthonormal system of L2[0,1] and is zero outside [0,1], by the Parseval identity, expression (6.27) can be written as the scalar product:
(6.34)α,βSp,qn,k=∫01(ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t))(ϕαp,q(t)+(α(t)-β(t))Γ(t)ψαp,q(t))dt,
which ends the proof of the lemma.Notice that we can further simplify expression (6.26):(6.35)α,βSp,qn,k=δp,qn,k+∫01(α(t)-β(t))2Γ(t)(ψαn,k(t)ψαp,q(t))dt+∫01(α(t)-β(t))Γ(t)(ϕαn,k(t)ψαp,q(t)+ϕαp,q(t)ψαn,k(t))dt.
We are now in a position to show that the operator α,βS-Id can be seen as the limit of the finite-dimensional operator α,βSN-IdξΩN, in the following sense.Theorem 6.6.
The operatorα,βS-Id:l2(ℝ)→l2(ℝ) is a trace class operator, whose trace is given by
(6.36)Tr⁡(Sα,β-Id)=∫01(α(t)-β(t))dt+∫01hα(t)fα(t)2(α(t)-β(t))2dt.We prove this essential point in AppendixF. The proof consists in showing that the operator α,βS-Id is isometric to a Hilbert-Schmidt operator whose trace can be computed straightforwardly.


### 6.2.4. The Girsanov Theorem

We now proceed to prove the Girsanov theorem by extending the domain of the quadratic form associated withα,βS-Id to the space ξΩ′, which can only be done in law.Theorem 6.7.
In the infinite-dimensional case, the Radon-Nikodym derivative ofPβ=PβX-1 with respect to Pα=PαX-1 reads
(6.37)dPβ(ω)dPα(ω)=exp⁡(-12(∫01β(t)-α(t)dt+Ξ(ω)T(α,βS-IdξΩ′)Ξ(ω))),
which in terms of the Itô stochastic integral reads
(6.38)dPβ(ω)dPα(ω)=exp⁡⁡(∫01β(t)-α(t)fα2(t)αXt(ω)gα(t)d(αXt(ω)gα(t)))-12∫01(β(t)-α(t))2fα2(t)(αXt(ω)gα(t))2dt.In order to demonstrate the Girsanov theorem from our geometrical point of view, we need to establish the following result.Lemma 6.8.
The positive definite quadratic form onl2(ℝ)×l2(ℝ) associated with operator α,βS-Id:l2(ℝ)→l2(ℝ) is well defined on ξΩ′. Moreover, for all ξΩ′,
(6.39)(ξ,(α,βS-IdξΩ′)(ξ))=2∫01α(t)-β(t)fα2(t)αXt(ξ)gα(t)∘d(αXt(ξ)gα(t))+∫01(α(t)-β(t))2fα2(t)(αXt(ξ)gα(t))2dt,
where αXt(ξ)=αΦ(ξ) and ∘ refers to the Stratonovich integral and the equality is true in law.Proof of Theorem6.7.
We start by writing the finite-dimensional Radon-Nikodym derivative(6.40)dPβ(ω)dPα(ω)=JNα,β⋅lim⁡N→∞exp⁡(-12(ΞN(ω)T(α,βSN-IdξΩN)ΞN(ω))).
By Proposition 6.3, we have
(6.41)Jα,β=lim⁡N→∞Jα,βN=12∫01(α(t)-β(t))dt.
If, as usual, Ξ denotes a recursively indexed infinite-dimensional vector  of independent variables with law 𝒩(0,1) and ΞN=ξPN∘Ξ, writing ξn,k=Ξn,k(ω), we have
(6.42)ΞN(ω)T(α,βSN-IdξΩN)ΞN(ω)=∑n=0N∑p=0N∑0≤k<2n-1∑0≤q<2p-1ξn,k[α,βSN-IdξΩN]p,qn,kξp,q.
We know that ξ is almost surely in ξΩ′, and, by Lemma 6.8, we also know that on ξΩ′×Ωξ′(6.43)lim⁡N→∞(ξ,(Sα,βN-IdξΩN)(ξ))=(ξ,(Sα,β-IdΩξ′)(ξ)),
so that we can effectively write the infinite-dimensional Radon-Nikodym derivative on ξΩ′ as the point-wise limit of the finite-dimensional one  on ξΩN through the projectors ξPN:
(6.44)dPβdPα(ω)=lim⁡N→∞dPβNdPαN(ω),
which directly yields formula (6.37).
The derivation of the Girsanov formula (6.40) from (6.37) comes from the relationship between the Stratonovich and Itô formulas  since the quadratic variation of αXt/gα(t) and (α(t)-β(t))/fα2(t)×αXt/gα(t) reads
(6.45)∫01E(∫0tfα(s)dWs,α(t)-β(t)fα2(t)∫0tfα(s)dWs)=∫01(α(t)-β(t))dt.
Therefore, the expression of the Radon-Nikodym derivative in Lemma 6.8 can be written in terms of the Itô integrals as
(6.46)(ξ,(α,βS-IdξΩ′)(ξ))=∫01(α(t)-β(t))dt+2∫01α(t)-β(t)f2(t)αXt(ξ)gα(t)d(αXt(ξ)gα(t))+∫01(α(t)-β(t))2f2(t)(αXt(ξ)gα(t))2dt
and the Radon-Nikodym derivative as
(6.47)dPβdPα(ω)=exp⁡⁡(∫01β(t)-α(t)fα2(t)αXt(ω)gα(t)d(αXt(ω)gα(t))-12∫01(β(t)-α(t))2fα2(t)(αXt(ω)gα(t))2dt).Observe that, ifα(t)=0, we recover the familiar expression(6.48)dPβdPα(ω)=exp⁡(∫01β(t)Γ(t)Wt(ω)dWt(ω)-12∫01β(t)2Γ(t)W(ω)t2dt).Conclusion and Perspectives
The discrete construction we present displays both analytical and numerical interests for further applications. From the analysis viewpoint, even if the basis does not exhibit the same orthogonal properties as the Karhunen-Loève decomposition, it has the important advantage of saving the structure of sample paths through its property of strong pathwise convergence and of providing a multiscale representation of the processes, which contrasts with the convergence in the mean of the Karhunen-Loève decomposition. From the numerical viewpoint, three Haar-like properties make our decomposition particularly suitable for certain numerical computations: (i) all basis elements have compact support on an open interval that has the structure of dyadic rational endpoints, (ii) these intervals are nested and become smaller for larger indices of the basis element, and (iii) for any interval endpoint, only a finite number of basis elements are nonzero at that point. Thus the expansion in our basis, when evaluated at an interval endpoint (e.g., dyadic rational), terminates in a finite number of steps. Moreover, the very nature of the construction based on an increasingly refined description of the sample paths paves the way to coarse-graining approaches similar to wavelet decompositions in signal processing. In view of this, our framework offers promising applications.Dichotomic Search of First-Hitting Times
The first application we envisage concerns the problem of first-hitting times. Because of its manifold applications, finding the time when a process first exits a given region  is a central question of stochastic calculus. However, closed-form theoretical results are scarce and one often has to resort to numerical algorithms [59]. In this regard, the multiresolution property suggests an exact scheme to simulate sample paths of a Gaussian Markov process X in an iterative “top-down’’ fashion. Assuming the intervals are dyadic rational  and that we have a conditional knowledge of a sample path on the dyadic points of DN={k2-N∣0≤k≤2N}, one can decide to further the simulation of this sample path at any time t in DN+1 by drawing a point according to the conditional law of Xt given {Xt}t∈DN, which is simply expressed in the framework of our construction. This property can be used for great advantages in numerical computations such as dichotomic search algorithms for first passage times: the key element is to find an estimate of the true conditional probability that a hitting time has occurred when knowing the value of the process at two given times, one in the past and one in the future. With such an estimate, an efficient strategy to look for passage times consists in refining the sample path when and only when its trajectory is estimated likely to actually cross the barrier. Thus the sample path of the process is represented at poor temporal resolution when it is far from the boundary and at increasingly higher resolution closer to the boundary. Such an algorithmic principle achieves a high level of precision in the computation of the first-hitting time, while demanding far less operation than usual stochastic Runge-Kutta scheme. This approach has been successfully implemented for the one-dimensional case [58], see Figure 4. In that article, the precision of the algorithm is controlled as well as the probability to evaluate a first hitting time substantially different from the actual value. The approach proves to be extremely efficient compared to customary methods. The general multidimensional approach proposed in the present paper allows direct generalization of these results to the computation of exit times in any dimension and for general smooth sets [30–32].Gaussian Deformation Modes in Nonlinear Diffusions
The present study is developed for the Gauss-Markov systems. However, many models arising in applied science present nonlinearities, and in that case, the construction based on a sum of Gaussian random variables will not generalize. However, the Gaussian case treated here can nevertheless be applied to perturbation of nonlinear differential equations with small noise. LetF:ℝ×ℝd↦ℝd be a nonlinear time-varying vector field, and let us assume that X0(t) is a stable (attractive) solution of the dynamical system:
(6.49)dXdt=F(t,X).
This function X0 can for instance be a fixed point (in which case it is a constant), a cycle (in which case it is periodic), or a general attractive orbit of the system. In the deterministic case, any solution having its initial condition in a given neighbourhood ℬ in ℝ×ℝd of the solution will asymptotically converge towards the solution, and therefore perturbations of the solutions are bounded. Let us now consider that the system is subject to a small amount of noise and define Y∈ℝd as the solution of the stochastic nonlinear differential equation:
(6.50)dYt=F(t,Yt)dt+ɛΓ(t,Yt)dWt.
Assuming that the noise is small (i.e., ɛ is a small parameter), because of the attractivity of the solution X0(t), the function Y(t) will remain very close to X0(t) (at least in a bounded time interval). In this region, we define ɛZt=Yt-X0(t). This stochastic variable is the solution of the equation
(6.51)dZt=1ɛ(F(t,X0(t)+ɛZt)-F(t,X0(t))+ɛΓ(t,X0(t)+ɛZt)dWt)=(∇xF)(t,X0(t))Zt+Γ(t,X0(t))dWt+O(ɛ).
The solution at the first order in ɛ is therefore the multidimensional Gaussian process with nonconstant coefficients:
(6.52)dZt=(∇xF)(t,X0(t))Zt+Γ(t,X0(t))dWt,
and our theory describes the solutions in a multiresolution framework. Notice that, in that perspective, our basis functions ψn,k can be seen as increasingly finer modes of deformation of a deterministic trajectory. This approach appears particularly relevant to the theory of weakly interconnected neural oscillators in computational neuroscience [60]. Indeed, one of the most popular approaches of this field, the phase model theory, formally consists in studying how perturbations are integrated in the neighborhood of an attracting cycle [61, 62].
All these instances are exemplary of how our multiresolution description of the Gauss-Markov processes offers a simple yet rigorous tool to broach a large number of open problems and promises fascinating applications both in theoretical and in applied science.Figure 4
First passage time for an Ornstein-Uhlenbeck processU with a elastic coefficient α=1 and a noise intensity Γ=1, the barrier is constant Λ=1 and the initial condition is U0=0. The plots represent at different scales a realization Ut(ω) for which the algorithm of [58] returns a first passage time τ(ω)=8.00469684 with a resolution δt=221=5107 after 5 recursive calls. The whole sample path is represented in (a) and a series of zooms is carried out around τ(ω) in (b), (c), and (d). The dilation coefficients are set according to the scale invariance of a Wiener process with the time scale being expanded by 20 and the distance scale by 45 during each dilation. The series (b), (c), and (c), (d) zoom on regions where the sample path gets close to the barrier. The simulation of the sample path has required 683 subdivisions and illustrates an unfavorable situation since the expected number of divisions is approximatively 284 for this particular setting.


## 6.2.1. Lift Operators

Depending on the space we  are considering (either coefficients or trajectories), we define the two following operators mapping the processαXt on βXt.(1)
Thecoefficients lift operator α,βG is the linear operator mapping in ξΩ′ the process αX on the process βX:
(6.10)α,βG=Δβ∘Ψα:(ξΩ′,B(ξΩ′))⟶(ξΩ′,B(ξΩ′)).
For any ξ∈ξΩ′, the operator α,βG maps a sample path of αX on a sample path of βX.(2)
Theprocess lift operator α,βF is the linear operator mapping in xΩ the process αX on the process βX:
(6.11)α,βH=αΨ∘αΔ:(xΩ,B(xΩ))⟶(xΩ,B(xΩ)).
We summarize the properties of these operators now.Proposition 6.3.
The operatorsα,βG and α,βH satisfy the following properties.(i)
They are linear measurable bijections(ii)
For everyN>0, the function α,βGN=PN∘α,βG∘IN:ξΩN′→ξΩN (resp., α,βHN=PN∘α,βH∘IN:ξΩN′→ξΩN) is a finite-dimensional linear operator, whose matrix representation is triangular in the natural basis of ξΩN (resp., ξΩN′) and whose eigenvalues α,βνn,k are given by
(6.12)α,βνn,k=gα(mn,k)gβ(mn,k)βMn,kαMn,k,0≤n≤N,0≤k<2N-1
(resp., β,ανn,k=(α,βνn,k)-1).(iii)
α,βG and α,βH are bounded operators for the spectral norm with
(6.13)‖α,βG‖2=sup⁡nsup⁡kνα,βn,k≤sup⁡gαinf⁡gβsup⁡fα2inf⁡fβ2<∞,
and ∥α,βH∥2=∥β,αG∥2<∞.(iv)
The determinants ofα,βGN (denoted by α,βJN) and α,βHN admit a limit when N tends to infinity:
(6.14)Jα,β=lim⁡N→∞Jα,βN=exp⁡(12(∫01(α(t)-β(t))dt)),lim⁡N→∞det⁡(Hα,βN)=exp⁡(12(∫01(β(t)-α(t))dt))=Jβ,α.The proof of these properties elementary stems from the analysis done on the functionsΨ and Δ that were previously performed, and these are detailed in Appendix C.


## 6.2.2. Radon-Nikodym Derivatives

From the properties proved on the lift operators, we are in position to further analyze the relationship between the probability distributions ofαX and βX. We first consider the finite-dimensional processes αXN and βXN. We emphasize that,  throughout this section, all equalities are true pathwise.Lemma 6.4.
Given the finite-dimensional measuresPαN and PβN, the Radon-Nikodym derivative of PβN with respect to PαN satisfies
(6.15)dPβNdPαN(ω)=Jα,βN⋅exp⁡(-12(ΞN(ω)T(Sα,βN-IdξΩN)ΞN(ω)))
with α,βSN=α,βGNT·α,βGN and the equality is true pathwise.Proof.
In the finite-dimensional case, for allN>0, we can write that PαN, PβNpαN, and the Lebesgue measure xΩN are mutually absolutely continuous: we denote by pαN and pβN the Gaussian density of PαN and PβN with respect to the Lebesgue measure on xΩN. Therefore, in the finite-dimensional case, the Radon-Nikodym derivative of PβN with respect to PαN is defined  to be pathwise and is simply given by the quotient of the density of the vector {βXN(mi,j)} with the density of the vector {αXN(mi,j)} for 0≤i≤N, 0≤j<2i-1, that is,
(6.16)dPβNdPαN(ω)=pβN(αXN(ω))pαN(αXN(ω))=det⁡(αΣN)det⁡(βΣN)⋅exp⁡(-12(αXN(ω)T(βΣN-1-αΣN-1)αXN(ω))).
We first make explicit
(6.17)det⁡(αΣN)det⁡(βΣN)=det⁡(αΣN⋅βΣN-1)=det⁡(αΨN⋅αΨNT⋅βΔNT⋅βΔN)=det⁡(βΔN⋅αΨN⋅αΨNT⋅βΔNT)=det⁡(α,βGN⋅α,βGNT)=det⁡(α,βGN)2.
Then, we rearrange the exponent using the Cholesky decomposition:
(6.18)ΣβN-1-ΣαN-1=ΔβNT⋅ΔβN-ΔαNT⋅ΔαN,
so that we write the exponent of (6.16) as
(6.19)αXNT(ΔβNT⋅ΔβN-ΔαNT⋅ΔαN)XαN=ΞNT⋅ΨαNT(ΔβNT⋅ΔβN-αΔNT⋅ΔαN)ΨαN(ω)⋅ΞN=ΞNT(Gα,βNT⋅Gα,βN-IdξΩN)ΞN.
We finally reformulate (6.16) as
(6.20)dPβNdPαN(ω)=Jα,βN⋅exp⁡(-12(ΞN(ω)T(α,βGNT⋅α,βGN-IdξΩN)ΞN(ω))).Let us now justify from a geometrical point of view why this formula is a direct consequence of the finite-dimensional change of variable formula on the model spaceξΩN. If we introduce αΔN, the coefficient application related to αXN, we know that ΞN=αΔN(αXN) follows a normal law 𝒩(0,IξΩN). We denote by pξN its standard Gaussian density with respect to the Lebesgue measure on ξΩN. We also know that(6.21)αΔN(βXN)=(αΔN∘α,βFN)(αXN)=β,αGN(αΔN(αXN))=β,αGN(ΞN).Sinceβ,αGN is linear, the change of the variable formula directly entails that β,αGN(ΞN) admits on ξΩN(6.22)pβ,αN(ξN)=|det⁡(α,βG)|pξN[ξNT(α,βGNT⋅α,βGN)ξN]
as density with respect to the Lebesgue measure. Consider now B as a measurable set of (xΩN,ℬ(xΩN)); then we have(6.23)PβN(B)=∫αΔN(B)pβ,αN(ξN)dξN=∫αΔN(B)pβ,αN(ξN)pξN(ξN)pξN(ξN)dξN=∫Bpβ,αN(αΔN(XN))pξN(αΔN(XN))dPαN(XN),
from which we immediately conclude.


## 6.2.3. The Trace Class Operator

The pathwise expression of the Radon-Nikodym derivative extends to the infinite-dimensional representation ofαX and βX. This extension involves technical analysis on the infinite-dimensional Hilbert space l2(ℝ). We have shown in Proposition 6.3 that the application α,βG was bounded for the spectral norm. Therefore, we have, for any ξ in l2(ℝ), the inequality(6.24)‖Gα,β(ξ)‖2≤‖Gα,β‖2⋅‖ξ‖2
implying that α,βG maps l2(ℝ) into l2(ℝ). We can then define the adjoint operator α,βGT from l2(ℝ) to l2(ℝ), which is given by(6.25)∀ξ,η∈l2(R),(Gα,βT(η),ξ)=(η,Gα,β(ξ)).
Let us now consider the self-adjoint operator α,βS=Gα,βT∘Gα,β:l2(ℝ)→l2(ℝ). This operator is the infinite-dimensional counterpart of the matrix α,βSN.Lemma 6.5.
Considering the coefficients of the matrix representation ofSα,β in the natural basis en,k of l2(ℝ) given as α,βSp,qn,k=(en,k,Sα,β(ep,q)), one has
(6.26)Sα,βp,qn,k=∫01(ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t))(ϕαp,q(t)+(α(t)-β(t))Γ(t)ψαp,q(t))dt.Proof.
Assume thatmax(n,p)≤N and that (n,k) and (p,q)∈ℐN. With the notations used previously with U, an open neighbourhood of [0,1], we have
(6.27)α,βSp,qn,k=(α,βG(en,k),α,βG(ep,q))=∑(i,j)∈I∫Uδβi,j(t)ψαn,k(t)dt∫Uδβi,j(t)ψαp,q(t)dt.
Since we have by definition αψn,k=𝒦α[αϕn,k], δβi,j=𝒟β[ϕβi,j],   𝒦α-1=𝒟α*,   𝒟β-1=𝒦β* on the space D0′(U), we have
(6.28)βϕi,j(t)=Kβ*[δβi,j](t)=fβ(t)∫Ugβ(s)δβi,j(s)ds,(6.29)αϕn,k(t)=Dα*[αψn,k](t)=1fα(t)ddt(αψn,k(t)gα(t)).
From there using (6.28), we can write the integration by part formula in the sense of the generalized functions to get
(6.30)∫Uδβi,j(t)αψn,k(t)dt=∫U1fβ(t)ddt(αψn,k(t)gβ(t))ϕβi,j(t)dt=∫UDβ*[ψαn,k](t)ϕβi,j(t)dt.
We now compute using (6.29):
(6.31)Dβ*[ψαn,k](t)=1fβ(t)ddt(ψαn,k(t)gα(t)gα(t)gβ(t))=gα(t)f(t)gβ(t)fβ(t)ϕα(t)-ddt(gα(t)gβ(t))ψαn,k(t)gβ(t)fβ(t).
Specifying gα, gβ, fα, and fβ and recalling the relations
(6.32)gα(t)fα(t)=gβ(t)fβ(t)=Γ(t),ddt(gα(t)gβ(t))=(α(t)-β(t))gα(t)gβ(t),
we rewrite the function (6.31) in L2[0,1] as
(6.33)βD*[ψαn,k](t)=1fβ(t)ddt(αψn,k(t)gβ(t))=ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t).
Now, since the family βϕn,k forms a complete orthonormal system of L2[0,1] and is zero outside [0,1], by the Parseval identity, expression (6.27) can be written as the scalar product:
(6.34)α,βSp,qn,k=∫01(ϕαn,k(t)+(α(t)-β(t))Γ(t)ψαn,k(t))(ϕαp,q(t)+(α(t)-β(t))Γ(t)ψαp,q(t))dt,
which ends the proof of the lemma.Notice that we can further simplify expression (6.26):(6.35)α,βSp,qn,k=δp,qn,k+∫01(α(t)-β(t))2Γ(t)(ψαn,k(t)ψαp,q(t))dt+∫01(α(t)-β(t))Γ(t)(ϕαn,k(t)ψαp,q(t)+ϕαp,q(t)ψαn,k(t))dt.
We are now in a position to show that the operator α,βS-Id can be seen as the limit of the finite-dimensional operator α,βSN-IdξΩN, in the following sense.Theorem 6.6.
The operatorα,βS-Id:l2(ℝ)→l2(ℝ) is a trace class operator, whose trace is given by
(6.36)Tr⁡(Sα,β-Id)=∫01(α(t)-β(t))dt+∫01hα(t)fα(t)2(α(t)-β(t))2dt.We prove this essential point in AppendixF. The proof consists in showing that the operator α,βS-Id is isometric to a Hilbert-Schmidt operator whose trace can be computed straightforwardly.


## 6.2.4. The Girsanov Theorem

We now proceed to prove the Girsanov theorem by extending the domain of the quadratic form associated withα,βS-Id to the space ξΩ′, which can only be done in law.Theorem 6.7.
In the infinite-dimensional case, the Radon-Nikodym derivative ofPβ=PβX-1 with respect to Pα=PαX-1 reads
(6.37)dPβ(ω)dPα(ω)=exp⁡(-12(∫01β(t)-α(t)dt+Ξ(ω)T(α,βS-IdξΩ′)Ξ(ω))),
which in terms of the Itô stochastic integral reads
(6.38)dPβ(ω)dPα(ω)=exp⁡⁡(∫01β(t)-α(t)fα2(t)αXt(ω)gα(t)d(αXt(ω)gα(t)))-12∫01(β(t)-α(t))2fα2(t)(αXt(ω)gα(t))2dt.In order to demonstrate the Girsanov theorem from our geometrical point of view, we need to establish the following result.Lemma 6.8.
The positive definite quadratic form onl2(ℝ)×l2(ℝ) associated with operator α,βS-Id:l2(ℝ)→l2(ℝ) is well defined on ξΩ′. Moreover, for all ξΩ′,
(6.39)(ξ,(α,βS-IdξΩ′)(ξ))=2∫01α(t)-β(t)fα2(t)αXt(ξ)gα(t)∘d(αXt(ξ)gα(t))+∫01(α(t)-β(t))2fα2(t)(αXt(ξ)gα(t))2dt,
where αXt(ξ)=αΦ(ξ) and ∘ refers to the Stratonovich integral and the equality is true in law.Proof of Theorem6.7.
We start by writing the finite-dimensional Radon-Nikodym derivative(6.40)dPβ(ω)dPα(ω)=JNα,β⋅lim⁡N→∞exp⁡(-12(ΞN(ω)T(α,βSN-IdξΩN)ΞN(ω))).
By Proposition 6.3, we have
(6.41)Jα,β=lim⁡N→∞Jα,βN=12∫01(α(t)-β(t))dt.
If, as usual, Ξ denotes a recursively indexed infinite-dimensional vector  of independent variables with law 𝒩(0,1) and ΞN=ξPN∘Ξ, writing ξn,k=Ξn,k(ω), we have
(6.42)ΞN(ω)T(α,βSN-IdξΩN)ΞN(ω)=∑n=0N∑p=0N∑0≤k<2n-1∑0≤q<2p-1ξn,k[α,βSN-IdξΩN]p,qn,kξp,q.
We know that ξ is almost surely in ξΩ′, and, by Lemma 6.8, we also know that on ξΩ′×Ωξ′(6.43)lim⁡N→∞(ξ,(Sα,βN-IdξΩN)(ξ))=(ξ,(Sα,β-IdΩξ′)(ξ)),
so that we can effectively write the infinite-dimensional Radon-Nikodym derivative on ξΩ′ as the point-wise limit of the finite-dimensional one  on ξΩN through the projectors ξPN:
(6.44)dPβdPα(ω)=lim⁡N→∞dPβNdPαN(ω),
which directly yields formula (6.37).
The derivation of the Girsanov formula (6.40) from (6.37) comes from the relationship between the Stratonovich and Itô formulas  since the quadratic variation of αXt/gα(t) and (α(t)-β(t))/fα2(t)×αXt/gα(t) reads
(6.45)∫01E(∫0tfα(s)dWs,α(t)-β(t)fα2(t)∫0tfα(s)dWs)=∫01(α(t)-β(t))dt.
Therefore, the expression of the Radon-Nikodym derivative in Lemma 6.8 can be written in terms of the Itô integrals as
(6.46)(ξ,(α,βS-IdξΩ′)(ξ))=∫01(α(t)-β(t))dt+2∫01α(t)-β(t)f2(t)αXt(ξ)gα(t)d(αXt(ξ)gα(t))+∫01(α(t)-β(t))2f2(t)(αXt(ξ)gα(t))2dt
and the Radon-Nikodym derivative as
(6.47)dPβdPα(ω)=exp⁡⁡(∫01β(t)-α(t)fα2(t)αXt(ω)gα(t)d(αXt(ω)gα(t))-12∫01(β(t)-α(t))2fα2(t)(αXt(ω)gα(t))2dt).Observe that, ifα(t)=0, we recover the familiar expression(6.48)dPβdPα(ω)=exp⁡(∫01β(t)Γ(t)Wt(ω)dWt(ω)-12∫01β(t)2Γ(t)W(ω)t2dt).Conclusion and Perspectives
The discrete construction we present displays both analytical and numerical interests for further applications. From the analysis viewpoint, even if the basis does not exhibit the same orthogonal properties as the Karhunen-Loève decomposition, it has the important advantage of saving the structure of sample paths through its property of strong pathwise convergence and of providing a multiscale representation of the processes, which contrasts with the convergence in the mean of the Karhunen-Loève decomposition. From the numerical viewpoint, three Haar-like properties make our decomposition particularly suitable for certain numerical computations: (i) all basis elements have compact support on an open interval that has the structure of dyadic rational endpoints, (ii) these intervals are nested and become smaller for larger indices of the basis element, and (iii) for any interval endpoint, only a finite number of basis elements are nonzero at that point. Thus the expansion in our basis, when evaluated at an interval endpoint (e.g., dyadic rational), terminates in a finite number of steps. Moreover, the very nature of the construction based on an increasingly refined description of the sample paths paves the way to coarse-graining approaches similar to wavelet decompositions in signal processing. In view of this, our framework offers promising applications.Dichotomic Search of First-Hitting Times
The first application we envisage concerns the problem of first-hitting times. Because of its manifold applications, finding the time when a process first exits a given region  is a central question of stochastic calculus. However, closed-form theoretical results are scarce and one often has to resort to numerical algorithms [59]. In this regard, the multiresolution property suggests an exact scheme to simulate sample paths of a Gaussian Markov process X in an iterative “top-down’’ fashion. Assuming the intervals are dyadic rational  and that we have a conditional knowledge of a sample path on the dyadic points of DN={k2-N∣0≤k≤2N}, one can decide to further the simulation of this sample path at any time t in DN+1 by drawing a point according to the conditional law of Xt given {Xt}t∈DN, which is simply expressed in the framework of our construction. This property can be used for great advantages in numerical computations such as dichotomic search algorithms for first passage times: the key element is to find an estimate of the true conditional probability that a hitting time has occurred when knowing the value of the process at two given times, one in the past and one in the future. With such an estimate, an efficient strategy to look for passage times consists in refining the sample path when and only when its trajectory is estimated likely to actually cross the barrier. Thus the sample path of the process is represented at poor temporal resolution when it is far from the boundary and at increasingly higher resolution closer to the boundary. Such an algorithmic principle achieves a high level of precision in the computation of the first-hitting time, while demanding far less operation than usual stochastic Runge-Kutta scheme. This approach has been successfully implemented for the one-dimensional case [58], see Figure 4. In that article, the precision of the algorithm is controlled as well as the probability to evaluate a first hitting time substantially different from the actual value. The approach proves to be extremely efficient compared to customary methods. The general multidimensional approach proposed in the present paper allows direct generalization of these results to the computation of exit times in any dimension and for general smooth sets [30–32].Gaussian Deformation Modes in Nonlinear Diffusions
The present study is developed for the Gauss-Markov systems. However, many models arising in applied science present nonlinearities, and in that case, the construction based on a sum of Gaussian random variables will not generalize. However, the Gaussian case treated here can nevertheless be applied to perturbation of nonlinear differential equations with small noise. LetF:ℝ×ℝd↦ℝd be a nonlinear time-varying vector field, and let us assume that X0(t) is a stable (attractive) solution of the dynamical system:
(6.49)dXdt=F(t,X).
This function X0 can for instance be a fixed point (in which case it is a constant), a cycle (in which case it is periodic), or a general attractive orbit of the system. In the deterministic case, any solution having its initial condition in a given neighbourhood ℬ in ℝ×ℝd of the solution will asymptotically converge towards the solution, and therefore perturbations of the solutions are bounded. Let us now consider that the system is subject to a small amount of noise and define Y∈ℝd as the solution of the stochastic nonlinear differential equation:
(6.50)dYt=F(t,Yt)dt+ɛΓ(t,Yt)dWt.
Assuming that the noise is small (i.e., ɛ is a small parameter), because of the attractivity of the solution X0(t), the function Y(t) will remain very close to X0(t) (at least in a bounded time interval). In this region, we define ɛZt=Yt-X0(t). This stochastic variable is the solution of the equation
(6.51)dZt=1ɛ(F(t,X0(t)+ɛZt)-F(t,X0(t))+ɛΓ(t,X0(t)+ɛZt)dWt)=(∇xF)(t,X0(t))Zt+Γ(t,X0(t))dWt+O(ɛ).
The solution at the first order in ɛ is therefore the multidimensional Gaussian process with nonconstant coefficients:
(6.52)dZt=(∇xF)(t,X0(t))Zt+Γ(t,X0(t))dWt,
and our theory describes the solutions in a multiresolution framework. Notice that, in that perspective, our basis functions ψn,k can be seen as increasingly finer modes of deformation of a deterministic trajectory. This approach appears particularly relevant to the theory of weakly interconnected neural oscillators in computational neuroscience [60]. Indeed, one of the most popular approaches of this field, the phase model theory, formally consists in studying how perturbations are integrated in the neighborhood of an attracting cycle [61, 62].
All these instances are exemplary of how our multiresolution description of the Gauss-Markov processes offers a simple yet rigorous tool to broach a large number of open problems and promises fascinating applications both in theoretical and in applied science.Figure 4
First passage time for an Ornstein-Uhlenbeck processU with a elastic coefficient α=1 and a noise intensity Γ=1, the barrier is constant Λ=1 and the initial condition is U0=0. The plots represent at different scales a realization Ut(ω) for which the algorithm of [58] returns a first passage time τ(ω)=8.00469684 with a resolution δt=221=5107 after 5 recursive calls. The whole sample path is represented in (a) and a series of zooms is carried out around τ(ω) in (b), (c), and (d). The dilation coefficients are set according to the scale invariance of a Wiener process with the time scale being expanded by 20 and the distance scale by 45 during each dilation. The series (b), (c), and (c), (d) zoom on regions where the sample path gets close to the barrier. The simulation of the sample path has required 683 subdivisions and illustrates an unfavorable situation since the expected number of divisions is approximatively 284 for this particular setting.



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