# Time Reversal of Volterra Processes Driven Stochastic Differential Equations

Authors: L. Decreusefond
Journal: International Journal of Stochastic Analysis
 (2013)

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## Abstract

We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past-dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.


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## Body

## 1. Introduction

Fractional Brownian motion (fBm for short) of Hurst indexH∈[0,1] is the Gaussian process which admits the following representation: for any t≥0,
(1)BH(t)=∫0t‍KH(t,s)dB(s),
where B is a one-dimensional Brownian motion and KH is a triangular kernel, that is, KH(t,s)=0 for s>t, the definition of which is given in (46). Fractional Brownian motion is probably the first process which is not a semimartingale and for which it is still interesting to develop a stochastic calculus. That means we want to define a stochastic integral and solve stochastic differential equations driven by such a process. From the very beginning of this program, two approaches do exist. One approach is based on the Hölder continuity or the finite p variation of the fBm sample paths. The other way to proceed relies on the gaussianity of fBm. The former is mainly deterministic and was initiated by Zähle [1], Feyel and de la Pradelle [2], and Russo and Vallois [3, 4]. Then, came the notion of rough paths was introduced by Lyons [5], whose application to fBm relies on the work of Coutin and Qian [6]. These works have been extended in the subsequent works [7–17]. A new way of thinking came with the independent but related works of Feyel, de la Pradelle [18], and Gubinelli [19]. The integral with respect to fBm was shown to exist as the unique process satisfying some characterization (analytic in the case of [18], algebraic in [19]). As a byproduct, this showed that almost all the existing integrals throughout the literature were all the same as they all satisfy these two conditions. Behind each approach, but the last too, is a construction of an integral defined for a regularization of fBm, then the whole work is to show that, under some convenient hypothesis, the approximate integrals converge to a quantity which is called the stochastic integral with respect to fBm. The main tool to prove the convergence is either integration by parts in the sense of fractional deterministic calculus, or enrichment of the fBm by some iterated integrals proved to exist independently or by analytic continuation [20, 21].In the probabilistic approach [22–30], the idea is also to define an approximate integral and then prove its convergence. It turns out that the key tool is here the integration by parts in the sense of Malliavin calculus.In dimension greater than one, with the deterministic approach, one knows how to define the stochastic integral and prove existence and uniqueness of fBm-driven SDEs for fBm with Hurst index greater than1/4. Within the probabilistic framework, one knows how to define a stochastic integral for any value of H but one cannot prove existence and uniqueness of SDEs whatever the value of H. The primary motivation of this work is to circumvent this problem.In [26, 27], we defined stochastic integrals with respect to fBm as a “damped-Stratonovitch” integral with respect to the underlying standard Brownian motion. This integral is defined as the limit of Riemann-Stratonovitch sums, the convergence of which is proved after an integration by parts in the sense of Malliavin calculus. Unfortunately, this manipulation generates nonadaptiveness: formally the result can be expressed as
(2)∫0t‍u(s)∘dBH(s)=δ(𝒦t*u)+trace(𝒦t*∇u),
where 𝒦 is defined by
(3)𝒦f(t)=ddt∫0t‍KH(t,s)f(s)ds
and 𝒦t* is the adjoint of 𝒦 in ℒ2([0,t],R). In particular, there exists k such that
(4)𝒦t*f(s)=∫st‍k(t,u)f(u)du
for any f∈ℒ2([0,t],R) so that even if u is adapted (with respect to the Brownian filtration), the process (s↦𝒦t*u(s)) is anticipative. However, the stochastic integral process (t↦∫0t‍u(s)∘dBH(s)) remains adapted; hence, the anticipative aspect is, in some sense, artificial. The motivation of this work is to show that, up to time reversal, we can work with adapted process and Itô integrals. The time-reversal properties of fBm were already studied in [31] in a different context. It was shown there that the time reversal of the solution of an fBm-driven SDE of the form
(5)dY(t)=u(Y(t))dt+dBH(t)
is still a process of the same form. With a slight adaptation of our method to fBm-driven SDEs with drift, one should recover the main theorem of [31].In what follows, there is no restriction on the dimension, but we need to assume that any component ofBH is an fBm of Hurst index greater than 1/2. Consider that we want to solve the following equation:
(6)Xt=x+∫0t‍σ(Xs)∘dBH(s),0≤t≤T,
where σ is a deterministic function whose properties will be fixed below. It turns out that it is essential to investigate the more general equations:
(A′)Xr,t=x+∫rt‍σ(Xr,s)∘dBH(s),0≤r≤t≤T.
The strategy is then as follows. We will first consider the reciprocal problem:
(B′)Yr,t=x-∫rt‍σ(Ys,t)∘dBH(s),0≤r≤t≤T.The first critical point is that when we consider{Zr,t:=Yt-r,t,r∈[0,t]}, this process solves an adapted, past-dependent, and stochastic differential equation with respect to a standard Brownian motion. Moreover, because KH is lower-triangular and sufficiently regular, the trace term vanishes in the equation defining Z. We have then reduced the problem to an SDE with coefficients dependent on the past, a problem which can be handled by the usual contraction methods. We do not claim that the results presented are new (for instance, see the brilliant monograph [32] for detailed results obtained via rough paths theory), but it seems interesting to have purely probabilistic methods which show that fBm driven SDEs do have strong solutions which are homeomorphisms. Moreover, the approach given here shows the irreducible difference between the case H<1/2 and H>1/2. The trace term only vanishes in the latter situation, so that such an SDE is merely a usual SDE with past-dependent coefficients. This representation may be fruitful, for instance, to analyze the support and prove the absolute continuity of solutions of (6).This paper is organized as follows. After some preliminaries on fractional Sobolev spaces, often called Besov-Liouville space, we address, in Section3, the problem of Malliavin calculus and time reversal. This part is interesting in its own since stochastic calculus of variations is a framework oblivious to time. Constructing such a notion of time is achieved using the notion of resolution of the identity as introduced in [33, 34]. We then introduce the second key ingredient which is the notion of strict causality or quasinilpotence; see [35] for a related application. In Section 4, we show that solving (B′) reduces to solve a past-dependent stochastic differential equation with respect to a standard Brownian motion; see (C) below. Then, we prove existence, uniqueness, and some properties of this equation. Technical lemmas are postponed to Section 5.


## 2. Besov-Liouville Spaces

LetT>0 be fix real number. For a measurable function f:[0,T]→Rn, we define τTf by
(7)τTf(s)=f(T-s)foranys∈[0,T].
For t∈[0,T], etf will represent the restriction of f to [0,t], that is, etf=f1[0,t]. For any linear map A, we denote by AT*, its adjoint in ℒ2([0,T];Rn). For η∈(0,1], the space of η-Hölder continuous functions on [0,T] is equipped with the norm:
(8)∥f∥Hol(η)=sup0<s<t<T|f(t)-f(s)||t-s|η+∥f∥∞.Its topological dual is denoted byHol(η)*. For f∈ℒ1([0,T];Rn;dt) (denoted by ℒ1 for short), the left and right fractional integrals of f are defined by
(9)(I0+γf)(x)=1Γ(γ)∫0x‍f(t)(x-t)γ-1dt,x≥0,(IT-γf)(x)=1Γ(γ)∫xT‍f(t)(t-x)γ-1dt,x≤T,
where γ>0 and I0+0=IT-0=Id. For any γ≥0, p,q≥1, any f∈ℒp and g∈ℒq where p-1+q-1≤γ, we have
(10)∫0T‍f(s)(I0+γg)(s)ds=∫0T‍(IT-γf)(s)g(s)ds.The Besov-Liouville spaceI0+γ(ℒp):=ℐγ,p+ is usually equipped with the norm:
(11)∥I0+γf∥ℐγ,p+=∥f∥ℒp.Analogously, the Besov-Liouville spaceIT-γ(ℒp):=ℐγ,p- is usually equipped with the norm:
(12)∥IT--γf∥ℐγ,p-=∥f∥ℒp.We then have the following continuity results (see [2, 36]):Proposition 1.
Consider the following.(i)
If0<γ<1,1<p<1/γ, then I0+γ is a bounded operator from ℒp into ℒq with q=p(1-γp)-1.(ii)
For any0<γ<1 and any p≥1,ℐγ,p+ is continuously embedded in Hol(γ-1/p) provided that γ-1/p>0.(iii)
For any0<γ<β<1,Hol(β) is compactly embedded in ℐγ,∞.(iv)
Forγp<1, the spaces ℐγ,p+ and ℐγ,p- are canonically isomorphic. We will thus use the notation ℐγ,p to denote any of these spaces.


## 3. Malliavin Calculus and Time Reversal

Our reference probability space isΩ=𝒞0([0,T],Rn), the space of Rn-valued, continuous functions, null at time 0. The Cameron-Martin space is denoted by H and is defined as H=I0+1(ℒ2([0,T])). In what follows, the space ℒ2([0,T]) is identified with its topological dual. We denote by κ the canonical embedding from H into Ω. The probability measure P on Ω is such that the canonical map W:ω↦(ω(t),t∈[0,T]) defines a standard n-dimensional Brownian motion. A mapping ϕ from Ω into some separable Hilbert space ℌ is called cylindrical if it is of the form ϕ(w)=∑i=1d‍fi(〈vi,1,w〉,…,〈vi,n,w〉)xi, where for each i,fi∈𝒞0∞(Rn,R) and (vi,j,j=1,…,n) is a sequence of Ω*. For such a function we define ∇Wϕ as
(13)∇Wϕ(w)=∑i,j=1‍∂jfi(〈vi,1,w〉,…,〈vi,n,w〉)v~i,j⊗xi,
where v~ is the image of v∈Ω* by the map (I0+1∘κ)*. From the quasi-invariance of the Wiener measure [37], it follows that ∇W is a closable operator on Lp(Ω;ℌ), p≥1, and we will denote its closure with the same notation. The powers of ∇W are defined by iterating this procedure. For p>1,k∈N, we denote by 𝔻p,k(ℌ) the completion of ℌ-valued cylindrical functions under the following norm:
(14)∥ϕ∥p,k=∑i=0k‍∥(∇W)iϕ∥Lp(Ω;ℌ⊗ℒp([0,1])⊗i).We denote by𝕃p,1 the space 𝔻p,1(ℒp([0,T];Rn)). The divergence, denoted as δW, is the adjoint of ∇W: v belongs to DompδW whenever, for any cylindrical ϕ,
(15)|E[∫0T‍vs∇sWϕds]|≤c∥ϕ∥Lp
and, for such a process v,
(16)E[∫0T‍vs∇sWϕds]=E[ϕδWv].We introduced the temporary notationW for standard Brownian motion to clarify the forthcoming distinction between a standard Brownian motion and its time reversal. Actually, the time reversal of a standard Brownian is also a standard Brownian motion, and thus, both of them “live” in the same Wiener space. We now precise how their respective Malliavin gradient and divergence are linked. Consider B=(B(t),t∈[0,T]) an n-dimensional standard Brownian motion and B˘T=(B(T)-B(T-t),t∈[0,T]) its time reversal. Consider the following map:
(17)ΘT:Ω→Ωω↦ω˘=ω(T)-τTω,
and the commutative diagram:
(18)Note thatΘT-1=ΘT since ω(0)=0. For a function f∈𝒞b∞(Rnk), we define the following:
(19)∇rf(ω(t1),…,ω(tk))=∑j=1k‍∂jf(ω(t1),…,ω(tk))1[0,tj](r),∇˘rf(ω˘(t1),…,ω˘(tk))=∑j=1k‍∂jf(ω˘(t1),…,ω˘(tk))1[0,tj](r).The operator∇=∇B (resp., ∇˘=∇B˘) is the Malliavin gradient associated with a standard Brownian motion (resp., its time reversal). Since
(20)f(ω˘(t1),…,ω˘(tk))=f(ω(T)-ω(T-t1),…,ω(T)-ω(T-tk)),
we can consider f(ω˘(t1),…,ω˘(tk)) as a cylindrical function with respect to the standard Brownian motion. As such its gradient is given by
(21)∇rf(ω˘(t1),…,ω˘(tk))=∑j=1k‍∂jf(ω˘(t1),…,ω˘(tk))1[T-tj,T](r).We thus have, for any cylindrical functionF,
(22)∇F∘ΘT(ω)=τT∇˘F(ω˘).SinceΘT*P=P and τT is continuous from ℒp into itself for any p, it is then easily shown that the spaces 𝔻p,k and 𝔻˘p,k (with obvious notations) coincide for any p,k and that (22) holds for any element of one of these spaces. Hence we have proved the following theorem.Theorem 2.
For anyp≥1 and any integer k, the spaces 𝔻p,k and 𝔻˘p,k coincide. For any F∈𝔻p,k  for some p,k,
(23)∇(F∘ΘT)=τT∇˘(F∘ΘT),Pa.s.By duality, an analog result follows for divergences.Theorem 3.
A processu belongs to the domain of δ if and only if τTu belongs to the domain of δ˘, and, then, the following equality holds:
(24)δ˘(u(ω˘))(ω˘)=δ(τTu(ω˘))(ω)=δ(τTu∘ΘT)(ω).Proof.
Foru∈ℒ2, for cylindrical F, we have on the one hand:
(25)E[F(ω˘)δ˘u(ω˘)]=E[(∇˘F(ω˘),u)ℒ2],
and on the other hand,
(26)E[(∇˘F(ω˘),u)ℒ2]=E[(τT∇F∘ΘT(ω),u)ℒ2]=E[(∇F∘ΘT(ω),τTu)ℒ2]=E[F∘ΘT(ω)δ(τTu)(ω)]=E[F(ω˘)δ(τTu)(ω)].
Since this is valid for any cylindrical F, (24) holds for u∈ℒ2. Now, for u in the domain of divergence (see [37, 38]),
(27)δu=∑i‍((u,hi)ℒ2δhi-(∇u,hi⊗hi)ℒ2⊗ℒ2),
where (hi,i∈N) is an orthonormal basis of ℒ2([0,T];Rn). Thus, we have
(28)δ˘(u(ω˘))(ω˘)=∑i‍((u(ω˘),hi)ℒ2δ˘hi(ω˘)-(∇˘u(ω˘),hi⊗hi)ℒ2⊗ℒ2)=∑i‍((u(ω˘),hi)ℒ2δ(τThi)(ω)-(∇u(ω˘),τThi⊗hi)ℒ2⊗ℒ2)=∑i‍((τTu(ω˘),τThi)ℒ2δ(τThi)(ω)-(∇τTu(ω˘),τThi⊗τThi)ℒ2⊗ℒ2),
where we have taken into account that τT is in an involution. Since (hi,i∈N) is an orthonormal basis of ℒ2([0,T];Rn), identity (24) is satisfied for any u in the domain of δ.


### 3.1. Causality and Quasinilpotence

In anticipative calculus, the notion of trace of an operator plays a crucial role, We refer to [39] for more details on trace.Definition 4.
LetV be a bounded map from ℒ2([0,T];Rn) into itself. The map V is said to be trace class, whenever for one CONB (hn,n≥1) of ℒ2([0,T];Rn),
(29)∑n≥1‍|(Vhn,hn)ℒ2|isfinite.
Then, the trace of V is defined by
(30)trace(V)=∑n≥1‍(Vhn,hn)ℒ2.
It is easily shown that the notion of trace does not depend on the choice of the CONB.Definition 5.
A familyE of projections (Eλ, λ∈[0,1]) in ℒ2([0,T];Rn) is called a resolution of the identity if it satisfies the conditions:(1)
E0=0 and E1=Id(2)
EλEμ=Eλ∧μ(3)
limμ↓λEμ=Eλ for any λ∈[0,1) and limμ↑1Eμ=Id.
For instance, the familyE=(eλT,λ∈[0,1]) is a resolution of the identity in ℒ2([0,T];Rn).Definition 6.
A partitionπ of [0,T] is a sequence {0=t0<t1<⋯<tn=T}. Its mesh is denoted by |π| and defined by |π|=supi|ti+1-ti|.
The causality plays a crucial role in what follows. The next definition is just the formalization in terms of operator of the intuitive notion of causality.Definition 7.
A continuous mapV from ℒ2([0,T];Rn) into itself is said to be E causal if and only if the following condition holds:
(31)EλVEλ=EλVforanyλ∈[0,1].
For instance, an operatorV in integral form Vf(t)=∫0T‍V(t,s)f(s)ds is causal if and only if V(t,s)=0 for s≥t, that is, computing Vf(t) needs only the knowledge of f up to time t and not after. Unfortunately, this notion of causality is insufficient for our purpose, and we are led to introduce the notion of strict causality as in [40].Definition 8.
LetV be a causal operator. It is a strictly causal operator, whenever for any ε>0, there exists a partition π of [0,T] such that, for any π′={0=t0<t1<⋯<tn=T}⊂π,
(32)∥(Eti+1-Eti)V(Eti+1-Eti)∥ℒ2<ε,fori=0,…,n-1.
Note carefully that the identity map is causal but not strictly causal. Indeed, ifV=Id, for any s<t,
(33)∥(Et-Es)V(Et-Es)∥ℒ2=∥Et-Es∥ℒ2=1
since Et-Es is a projection. However, if V is hyper-contractive, we have the following result.Lemma 9.
Assume the resolution of the identity to be eitherE=(eλT,λ∈[0,1]) or E=(Id-e(1-λ)T,λ∈[0,1]). If V is an E causal map continuous from ℒ2 into ℒp for some p>2 then V is strictly E causal.Proof.
Letπ be any partition of [0,T]. Assume that E=(eλT,λ∈[0,1]), and the very same proof works for the other mentioned resolution of the identity. According to Hölder formula, we have for any 0≤s<t≤T,
(34)∥(Et-Es)V(Et-Es)f∥ℒ2=∫st‍|V(f1(s,t])(u)|2du≤(t-s)1-2/p∥V(f1(s,t])∥ℒp/2≤c(t-s)1-2/p∥f∥ℒ2.
Then, for any ε>0, there exists η>0 such that |π|<η implies ∥(Eti+1-Eti)V(Eti+1-Eti)f∥ℒ2≤ε for any {0=t0<t1<⋯<tn=T}⊂π and any i=0,…,n-1.The importance of strict causality lies in the next theorem we borrow from [40].Theorem 10.
The set of strictly causal operators coincides with the set of quasinilpotent operators, that is, trace-class operators such thattrace(Vn)=0 for any integer n≥1.Moreover, we have the following stability theorem.Theorem 11.
The set of strictly causal operators is a two-sided ideal in the set of causal operators.Definition 12.
LetE be a resolution of the identity in ℒ2([0,T];Rn). Consider the filtration ℱE defined as
(35)ℱtE=σ{δW(Eλh),λ≤t,h∈ℒ2}.
An ℒ2-valued random variable u is said to be ℱE adapted if, for any h∈ℒ2, the real valued process 〈Eλu,h〉 is ℱE-adapted. We denote by 𝔻p,kE(ℌ) the set of ℱE adapted random variables belonging to 𝔻p,k(ℌ).IfE=(eλT,λ∈[0,1]), the notion of ℱE adapted processes coincides with the usual one for the Brownian filtration, and it is well known that a process u is adapted if and only if ∇rWu(s)=0 for r>s. This result can be generalized to any resolution of the identity.Theorem 13 (Proposition 3.1 of [33]).
Letu belongs to 𝕃p,1. Then u is ℱE adapted if and only if ∇Wu is E causal.We then have the following key theorem.Theorem 14.
Assume the resolution of the identity to beE=(eλT,λ∈[0,1]) either E=(Id-e(1-λ)T,λ∈[0,1]) and that V is an E-strictly causal continuous operator from ℒ2 into ℒp for some p>2. Let u be an element of 𝔻2,1E(ℒ2). Then, V∇Wu is of trace class and we have trace(V∇Wu)=0.Proof.
Sinceu is adapted, ∇Wu is E-causal. According to Theorem 11, V∇Wu is strictly causal and the result follows by Theorem 10.In what follows,E0 is the resolution of the identity in the Hilbert space ℒ2 defined by eλTf=f1[0,λT] and E˘0 is the resolution of the identity defined by e˘λTf=f1[(1-λ)T,T]. The filtrations ℱE0 and ℱE˘0 are defined accordingly. Next lemma is immediate when V is given in the form of Vf(t)=∫0t‍V(t,s)f(s)ds. Unfortunately such a representation as an integral operator is not always available. We give here an algebraic proof to emphasize the importance of causality.Lemma 15.
LetV be a map from ℒ2([0,T];Rn) into itself such that V is E0-causal. Let V* be the adjoint of V in ℒ2([0,T];Rn). Then, the map τTVT*τT is E˘0-causal.Proof.
This is a purely algebraic lemma once we have noticed that(36)τTer=(Id-eT-r)τTforany0≤r≤T.
For, it suffices to write
(37)τTerf(s)=f(T-s)1[0,r](T-s)=f(T-s)1[T-r,T](s)=(Id-eT-r)τTf(s),forany0≤s≤T.
We have to show that
(38)erτTVT*τTer=erτTVT*τTorequivalently(39)erτTVτTer=τTVτTer,
since er*=er and τT*=τT. Now, (37) yields
(40)erτTVτTer=τTVτTer-eT-rVτTer.
Use (37) again to obtain
(41)eT-rVτTer=eT-rV(Id-eT-r)τT=(eT-rV-eT-rVeT-r)τT=0,
since V is E-causal.


### 3.2. Stratonovitch Integrals

In what follows,η belongs to (0,1] and V is a linear operator. For any p≥2, we set the following.Hypothesis 1 (p,η).
The linear mapV is continuous from ℒp([0,T];Rn) into the Banach space Hol(η).Definition 16.
Assume that Hypothesis1 (p,η) holds. The Volterra process associated to V, denoted by WV, is defined by
(42)WV(t)=δW(V(1[0,t])),∀t∈[0,T].For any subdivisionπ of [0,T], that is, π={0=t0<t1<⋯<tn=T}, of mesh |π|, we consider the Stratonovitch sums:
(43)Rπ(t,u)=δW(∑ti∈π‍1θi∫ti∧tti+1∧t‍Vu(r)dr1[ti,ti+1))+∑ti∈π‍1θi∬[ti∧t,ti+1∧t]2V(∇rWu)(s)dsdr.Definition 17.
We say thatu is V-Stratonovitch integrable on [0,t] whenever the family Rπ(t,u), defined in (43), converges in probability as |π| goes to 0. In this case the limit will be denoted by ∫0t‍u(s)∘dWV(s).Example 18.
The first example is the so-called Lévy fractional Brownian motion of Hurst indexH>1/2 defined as
(44)1Γ(H+1/2)∫0t‍(t-s)H-1/2dBs=δ(IT-H-1/2(1[0,t])).
This amounts to say that V=IT-H-1/2. Thus Hypothesis 1  (p,H-1/2-1/p) holds provided that p(H-1/2)>1.Example 19.
The other classical example is the fractional Brownian motion with stationary increments of Hurst indexH>1/2, which can be written as
(45)∫0t‍KH(t,s)dB(s),
where
(46)KH(t,r)=(t-r)H-(1/2)Γ(H+(1/2))F(12-H,H-12,H+12,1-tr)×1[0,t)(r).
The Gauss hypergeometric function F(α,β,γ,z) (see [41]) is the analytic continuation on ℂ×ℂ×ℂ∖{-1,-2,…}×{z∈ℂ,Arg|1-z|<π} of the power series:
(47)∑k=0+∞‍(α)k(β)k(γ)kk!zk,(a)0=1,(a)k=Γ(a+k)Γ(a)=a(a+1)⋯(a+k-1).
We know from [36] that KH is an isomorphism from ℒp([0,1]) onto ℐH+1/2,p+ and
(48)KHf=I0+1xH-1/2I0+H-1/2x1/2-Hf.
Consider that 𝒦H=I0+-1∘KH. Then it is clear that
(49)∫0t‍KH(t,s)dB(s)=∫0t‍(𝒦H)T*(1[0,t])(s)dB(s);
hence we are in the framework of Definition 17 provided that we take V=(𝒦H)T*. Hypothesis 1  (p,H-1/2-1/p) is satisfied provided that p(H-1/2)>1.The next theorem then follows from [26].Theorem 20.
Assume that Hypothesis1 (p,η) holds. Assume that u belongs to 𝕃p,1. Then u is V-Stratonovitch integrable, and there exists a process which we denote by DWu such that DWu belongs to Lp(P⊗ds) and
(50)∫0T‍u(s)∘dWV(s)=δW(Vu)+∫0T‍DWu(s)ds.
The so-called “trace-term” satisfies the following estimate:
(51)E[∫0T‍|DWu(r)|pdr]≤cTpη∥u∥𝕃p,1p,
for some universal constant c. Moreover, for any r≤T, eru is V-Stratonovitch integrable and
(52)∫0r‍u(s)∘dWV(s)=∫0T‍(eru)(s)∘dWV(s)=δW(Veru)+∫0r‍DWu(s)ds,
and we have the maximal inequality:
(53)E[∥∫0.u(s)∘dWV(s)∥Hol(η)p]≤c∥u∥𝕃p,1p,
where c does not depend on u.The main result of this Section is the following theorem which states that the time reversal of a Stratonovitch integral is an adapted integral with respect to the time-reversed Brownian motion. Due to its length, its proof is postponed to Section5.1.Theorem 21.
Assume that Hypothesis1 (p,η) holds. Let u belong to 𝕃p,1 and let V˘T=τtVτT. Assume furthermore that V is E˘0-causal and that u˘=u∘ΘT-1 is ℱE˘0-adapted. Then,
(54)∫T-tT-r‍τTu(s)∘dWV(s)=∫rt‍V˘T(1[r,t]u˘)(s)dB˘T(s),0≤r≤t≤T,
where the last integral is an Itô integral with respect to the time reversed Brownian motion B˘T(s)=B(T)-B(T-s)=ΘT(B)(s).Remark 22.
Note that, at a formal level, we could have an easy proof of this theorem. For instance, consider the Lévy fBm, and a simple computation shows thatV˘T=I0+H-1/2 for any T. Thus, we are led to compute trace(I0+H-1/2∇u). If we had sufficient regularity, we could write
(55)trace(I0+H-1/2∇u)=∫0T‍∫0s‍(s-r)H-3/2∇su(r)drds=0,
since ∇su(r)=0 for s>r for u adapted. Obviously, there are many flaws in these lines of proof. The operator I0+H-1/2∇u is not regular enough for such an expression of the trace to be true. Even more, there is absolutely no reason for V˘T∇u to be a kernel operator so we cannot hope such a formula. These are the reasons that we need to work with operators and not with kernels.


## 3.1. Causality and Quasinilpotence

In anticipative calculus, the notion of trace of an operator plays a crucial role, We refer to [39] for more details on trace.Definition 4.
LetV be a bounded map from ℒ2([0,T];Rn) into itself. The map V is said to be trace class, whenever for one CONB (hn,n≥1) of ℒ2([0,T];Rn),
(29)∑n≥1‍|(Vhn,hn)ℒ2|isfinite.
Then, the trace of V is defined by
(30)trace(V)=∑n≥1‍(Vhn,hn)ℒ2.
It is easily shown that the notion of trace does not depend on the choice of the CONB.Definition 5.
A familyE of projections (Eλ, λ∈[0,1]) in ℒ2([0,T];Rn) is called a resolution of the identity if it satisfies the conditions:(1)
E0=0 and E1=Id(2)
EλEμ=Eλ∧μ(3)
limμ↓λEμ=Eλ for any λ∈[0,1) and limμ↑1Eμ=Id.
For instance, the familyE=(eλT,λ∈[0,1]) is a resolution of the identity in ℒ2([0,T];Rn).Definition 6.
A partitionπ of [0,T] is a sequence {0=t0<t1<⋯<tn=T}. Its mesh is denoted by |π| and defined by |π|=supi|ti+1-ti|.
The causality plays a crucial role in what follows. The next definition is just the formalization in terms of operator of the intuitive notion of causality.Definition 7.
A continuous mapV from ℒ2([0,T];Rn) into itself is said to be E causal if and only if the following condition holds:
(31)EλVEλ=EλVforanyλ∈[0,1].
For instance, an operatorV in integral form Vf(t)=∫0T‍V(t,s)f(s)ds is causal if and only if V(t,s)=0 for s≥t, that is, computing Vf(t) needs only the knowledge of f up to time t and not after. Unfortunately, this notion of causality is insufficient for our purpose, and we are led to introduce the notion of strict causality as in [40].Definition 8.
LetV be a causal operator. It is a strictly causal operator, whenever for any ε>0, there exists a partition π of [0,T] such that, for any π′={0=t0<t1<⋯<tn=T}⊂π,
(32)∥(Eti+1-Eti)V(Eti+1-Eti)∥ℒ2<ε,fori=0,…,n-1.
Note carefully that the identity map is causal but not strictly causal. Indeed, ifV=Id, for any s<t,
(33)∥(Et-Es)V(Et-Es)∥ℒ2=∥Et-Es∥ℒ2=1
since Et-Es is a projection. However, if V is hyper-contractive, we have the following result.Lemma 9.
Assume the resolution of the identity to be eitherE=(eλT,λ∈[0,1]) or E=(Id-e(1-λ)T,λ∈[0,1]). If V is an E causal map continuous from ℒ2 into ℒp for some p>2 then V is strictly E causal.Proof.
Letπ be any partition of [0,T]. Assume that E=(eλT,λ∈[0,1]), and the very same proof works for the other mentioned resolution of the identity. According to Hölder formula, we have for any 0≤s<t≤T,
(34)∥(Et-Es)V(Et-Es)f∥ℒ2=∫st‍|V(f1(s,t])(u)|2du≤(t-s)1-2/p∥V(f1(s,t])∥ℒp/2≤c(t-s)1-2/p∥f∥ℒ2.
Then, for any ε>0, there exists η>0 such that |π|<η implies ∥(Eti+1-Eti)V(Eti+1-Eti)f∥ℒ2≤ε for any {0=t0<t1<⋯<tn=T}⊂π and any i=0,…,n-1.The importance of strict causality lies in the next theorem we borrow from [40].Theorem 10.
The set of strictly causal operators coincides with the set of quasinilpotent operators, that is, trace-class operators such thattrace(Vn)=0 for any integer n≥1.Moreover, we have the following stability theorem.Theorem 11.
The set of strictly causal operators is a two-sided ideal in the set of causal operators.Definition 12.
LetE be a resolution of the identity in ℒ2([0,T];Rn). Consider the filtration ℱE defined as
(35)ℱtE=σ{δW(Eλh),λ≤t,h∈ℒ2}.
An ℒ2-valued random variable u is said to be ℱE adapted if, for any h∈ℒ2, the real valued process 〈Eλu,h〉 is ℱE-adapted. We denote by 𝔻p,kE(ℌ) the set of ℱE adapted random variables belonging to 𝔻p,k(ℌ).IfE=(eλT,λ∈[0,1]), the notion of ℱE adapted processes coincides with the usual one for the Brownian filtration, and it is well known that a process u is adapted if and only if ∇rWu(s)=0 for r>s. This result can be generalized to any resolution of the identity.Theorem 13 (Proposition 3.1 of [33]).
Letu belongs to 𝕃p,1. Then u is ℱE adapted if and only if ∇Wu is E causal.We then have the following key theorem.Theorem 14.
Assume the resolution of the identity to beE=(eλT,λ∈[0,1]) either E=(Id-e(1-λ)T,λ∈[0,1]) and that V is an E-strictly causal continuous operator from ℒ2 into ℒp for some p>2. Let u be an element of 𝔻2,1E(ℒ2). Then, V∇Wu is of trace class and we have trace(V∇Wu)=0.Proof.
Sinceu is adapted, ∇Wu is E-causal. According to Theorem 11, V∇Wu is strictly causal and the result follows by Theorem 10.In what follows,E0 is the resolution of the identity in the Hilbert space ℒ2 defined by eλTf=f1[0,λT] and E˘0 is the resolution of the identity defined by e˘λTf=f1[(1-λ)T,T]. The filtrations ℱE0 and ℱE˘0 are defined accordingly. Next lemma is immediate when V is given in the form of Vf(t)=∫0t‍V(t,s)f(s)ds. Unfortunately such a representation as an integral operator is not always available. We give here an algebraic proof to emphasize the importance of causality.Lemma 15.
LetV be a map from ℒ2([0,T];Rn) into itself such that V is E0-causal. Let V* be the adjoint of V in ℒ2([0,T];Rn). Then, the map τTVT*τT is E˘0-causal.Proof.
This is a purely algebraic lemma once we have noticed that(36)τTer=(Id-eT-r)τTforany0≤r≤T.
For, it suffices to write
(37)τTerf(s)=f(T-s)1[0,r](T-s)=f(T-s)1[T-r,T](s)=(Id-eT-r)τTf(s),forany0≤s≤T.
We have to show that
(38)erτTVT*τTer=erτTVT*τTorequivalently(39)erτTVτTer=τTVτTer,
since er*=er and τT*=τT. Now, (37) yields
(40)erτTVτTer=τTVτTer-eT-rVτTer.
Use (37) again to obtain
(41)eT-rVτTer=eT-rV(Id-eT-r)τT=(eT-rV-eT-rVeT-r)τT=0,
since V is E-causal.


## 3.2. Stratonovitch Integrals

In what follows,η belongs to (0,1] and V is a linear operator. For any p≥2, we set the following.Hypothesis 1 (p,η).
The linear mapV is continuous from ℒp([0,T];Rn) into the Banach space Hol(η).Definition 16.
Assume that Hypothesis1 (p,η) holds. The Volterra process associated to V, denoted by WV, is defined by
(42)WV(t)=δW(V(1[0,t])),∀t∈[0,T].For any subdivisionπ of [0,T], that is, π={0=t0<t1<⋯<tn=T}, of mesh |π|, we consider the Stratonovitch sums:
(43)Rπ(t,u)=δW(∑ti∈π‍1θi∫ti∧tti+1∧t‍Vu(r)dr1[ti,ti+1))+∑ti∈π‍1θi∬[ti∧t,ti+1∧t]2V(∇rWu)(s)dsdr.Definition 17.
We say thatu is V-Stratonovitch integrable on [0,t] whenever the family Rπ(t,u), defined in (43), converges in probability as |π| goes to 0. In this case the limit will be denoted by ∫0t‍u(s)∘dWV(s).Example 18.
The first example is the so-called Lévy fractional Brownian motion of Hurst indexH>1/2 defined as
(44)1Γ(H+1/2)∫0t‍(t-s)H-1/2dBs=δ(IT-H-1/2(1[0,t])).
This amounts to say that V=IT-H-1/2. Thus Hypothesis 1  (p,H-1/2-1/p) holds provided that p(H-1/2)>1.Example 19.
The other classical example is the fractional Brownian motion with stationary increments of Hurst indexH>1/2, which can be written as
(45)∫0t‍KH(t,s)dB(s),
where
(46)KH(t,r)=(t-r)H-(1/2)Γ(H+(1/2))F(12-H,H-12,H+12,1-tr)×1[0,t)(r).
The Gauss hypergeometric function F(α,β,γ,z) (see [41]) is the analytic continuation on ℂ×ℂ×ℂ∖{-1,-2,…}×{z∈ℂ,Arg|1-z|<π} of the power series:
(47)∑k=0+∞‍(α)k(β)k(γ)kk!zk,(a)0=1,(a)k=Γ(a+k)Γ(a)=a(a+1)⋯(a+k-1).
We know from [36] that KH is an isomorphism from ℒp([0,1]) onto ℐH+1/2,p+ and
(48)KHf=I0+1xH-1/2I0+H-1/2x1/2-Hf.
Consider that 𝒦H=I0+-1∘KH. Then it is clear that
(49)∫0t‍KH(t,s)dB(s)=∫0t‍(𝒦H)T*(1[0,t])(s)dB(s);
hence we are in the framework of Definition 17 provided that we take V=(𝒦H)T*. Hypothesis 1  (p,H-1/2-1/p) is satisfied provided that p(H-1/2)>1.The next theorem then follows from [26].Theorem 20.
Assume that Hypothesis1 (p,η) holds. Assume that u belongs to 𝕃p,1. Then u is V-Stratonovitch integrable, and there exists a process which we denote by DWu such that DWu belongs to Lp(P⊗ds) and
(50)∫0T‍u(s)∘dWV(s)=δW(Vu)+∫0T‍DWu(s)ds.
The so-called “trace-term” satisfies the following estimate:
(51)E[∫0T‍|DWu(r)|pdr]≤cTpη∥u∥𝕃p,1p,
for some universal constant c. Moreover, for any r≤T, eru is V-Stratonovitch integrable and
(52)∫0r‍u(s)∘dWV(s)=∫0T‍(eru)(s)∘dWV(s)=δW(Veru)+∫0r‍DWu(s)ds,
and we have the maximal inequality:
(53)E[∥∫0.u(s)∘dWV(s)∥Hol(η)p]≤c∥u∥𝕃p,1p,
where c does not depend on u.The main result of this Section is the following theorem which states that the time reversal of a Stratonovitch integral is an adapted integral with respect to the time-reversed Brownian motion. Due to its length, its proof is postponed to Section5.1.Theorem 21.
Assume that Hypothesis1 (p,η) holds. Let u belong to 𝕃p,1 and let V˘T=τtVτT. Assume furthermore that V is E˘0-causal and that u˘=u∘ΘT-1 is ℱE˘0-adapted. Then,
(54)∫T-tT-r‍τTu(s)∘dWV(s)=∫rt‍V˘T(1[r,t]u˘)(s)dB˘T(s),0≤r≤t≤T,
where the last integral is an Itô integral with respect to the time reversed Brownian motion B˘T(s)=B(T)-B(T-s)=ΘT(B)(s).Remark 22.
Note that, at a formal level, we could have an easy proof of this theorem. For instance, consider the Lévy fBm, and a simple computation shows thatV˘T=I0+H-1/2 for any T. Thus, we are led to compute trace(I0+H-1/2∇u). If we had sufficient regularity, we could write
(55)trace(I0+H-1/2∇u)=∫0T‍∫0s‍(s-r)H-3/2∇su(r)drds=0,
since ∇su(r)=0 for s>r for u adapted. Obviously, there are many flaws in these lines of proof. The operator I0+H-1/2∇u is not regular enough for such an expression of the trace to be true. Even more, there is absolutely no reason for V˘T∇u to be a kernel operator so we cannot hope such a formula. These are the reasons that we need to work with operators and not with kernels.


## 4. Volterra-Driven SDEs

Let𝔊 be the group of homeomorphisms of Rn equipped with the distance. We introduce a distance d on 𝔊 by
(56)d(φ,ϕ)=ρ(φ,ϕ)+ρ(φ-1,ϕ-1),
where
(57)ρ(φ,ϕ)=∑N=1∞‍2-Nsup|x|≤N|φ(x)-ϕ(x)|1+sup|x|≤N|φ(x)-ϕ(x)|·
Then, 𝔊 is a complete topological group. Consider the equations:
(A)Xr,t=x+∫rt‍σ(Xr,s)∘dWV(s),0≤r≤t≤T,(B)Yr,t=x-∫rt‍σ(Ys,t)∘dWV(s),0≤r≤t≤T.
As a solution of (A) is to be constructed by “inverting” a solution of (B), we need to add to the definition of a solution of (A) or (B) the requirement of being a flow of homeomorphisms. This is the meaning of the following definition.Definition 23.
By a solution of(A), we mean a measurable map:
(58)Ω×[0,T]×[0,T]→𝔊(ω,r,t)↦(x↦Xr,t(ω,x))
such that the following properties are satisfied.(1)
For any0≤r≤t≤T, for any x∈Rn, Xr,t(ω,x) is σ{WV(s),r≤s≤t}-measurable.(2)
For any0≤r≤T, for any x∈Rn, the processes (ω,t)↦Xr,t(ω,x) and (ω,t)↦Xr,t-1(ω,x) belong to 𝕃p,1 for some p≥2.(3)
For any0≤r≤s≤t, for any x∈Rn, the following identity is satisfied:
(59)Xr,t(ω,x)=Xs,t(ω,Xr,s(ω,x)).(4)
Equation(A) is satisfied for any 0≤r≤t≤TP-a.s.Definition 24.
By a solution of(B), we mean a measurable map:
(60)Ω×[0,T]×[0,T]→𝔊(ω,r,t)↦(x↦Yr,t(ω,x))
such that the following properties are satisfied. (1)
For any0≤r≤t≤T, for any x∈Rn, Yr,t(ω,x) is σ{WV(s),r≤s≤t} measurable.(2)
For any0≤r≤T, for any x∈Rn, the processes (ω,r)↦Yr,t(ω,x) and (ω,r)↦Yr,t-1(ω,x) belong to 𝕃p,1 for some p≥2.(3)
Equation(B) is satisfied for any 0≤r≤t≤TP-a.s..(4)
For any0≤r≤s≤t, for any x∈Rn, the following identity is satisfied:
(61)Yr,t(ω,x)=Yr,s(ω,Ys,t(ω,x)).At last consider the equation, for any0≤r≤t≤T,
(C)Zr,t=x-∫rt‍V˘T(σ∘Z.,t1[r,t])(s)dB˘T(s),
where B is a standard n-dimensional Brownian motion.Definition 25.
By a solution of(C), we mean a measurable map:
(62)Ω×[0,T]×[0,T]→𝔊(ω,r,t)↦(x↦Zr,t(ω,x))
such that the following properties are satisfied.(1)
For any0≤r≤t≤T, for any x∈Rn, Zr,t(ω,x) is σ{B˘T(s),s≤r≤t} measurable.(2)
For any0≤r≤t≤T, for any x∈Rn, the processes (ω,r)↦Zr,t(ω,x) and (ω,r)↦Zr,t-1(ω,x) belong to 𝕃p,1 for some p≥2.(3)
Equation(C) is satisfied for any 0≤r≤t≤TP-a.s..Theorem 26.
Assume thatV˘T is an E0 causal map continuous from ℒp into ℐα,p for α>0 and p≥4 such that αp>1. Assume that σ is Lipschitz continuous and sublinear; see (96) for the definition. Then, there exists a unique solution to (C). Let Z denote this solution. For any (r,r′),
(63)E[|Zr,T-Zr′,T|p]≤c|r-r′|pη.
Moreover,
(64)(ω,r)↦Zr,s(ω,Zs,t(ω,x))∈𝕃p,1,foranyr≤s≤t≤T.Since this proof needs several lemmas, we defer it to Section5.2.Theorem 27.
Assume thatV˘T is an E0-causal map continuous from ℒp into ℐα,p for α>0 and p≥2 such that αp>1. For fixed T, there exists a bijection between the space of solutions of (B) on [0,T] and the set of solutions of (C).Proof.
Set(65)Zr,T(ω˘,x)=YT-r,T(ΘT-1(ω˘),x)
or equivalently
(66)Yr,T(ω,x)=ZT-r,T(ΘT(ω),x).
According to Theorem 21, Y satisfies (B) if and only if Z satisfies (C). The regularity properties are immediate since ℒp is stable by τT.The first part of the next result is then immediate.Corollary 28.
Assume thatV˘T is an E0-causal map continuous from ℒp into ℐα,p for α>0 and p≥2 such that αp>1. Then (B) has one and only one solution and for any 0≤r≤s≤t, for any x∈Rn, the following identity is satisfied:
(67)Yr,t(ω,x)=Yr,s(ω,Ys,t(ω,x)).Proof.
According to Theorems27 and 26,  (B) has at most one solution since (C) has a unique solution. As to the existence, points from (1) to (3) are immediately deduced from the corresponding properties of Z and (66).
According to Theorem26, (ω,r)↦Yr,s(ω,Ys,t(ω,x)) belongs to 𝕃p,1; hence, we can apply the substitution formula and
(68)Yr,s(ω,Ys,t(ω,x))=Ys,t(ω,x)-∫rs‍σ(Yτ,s(ω,x))∘dWV(τ)|x=Ys,t(ω,x)=x-∫st‍σ(Yτ,t(ω,x)∘dWV(τ)-∫rs‍σ(Yτ,s(ω,Ys,t(ω,x))dWV(τ))∘dWV(τ).
Set
(69)Rτ,t={Yτ,t(ω,x)fors≤τ≤tYτ,s(ω,Ys,t(ω,x))forr≤τ≤s.
Then, in view of (68), R appears to be the unique solution (B) and thus Rs,t(ω,x)=Ys,t(ω,x). Point (4) is thus proved.Corollary 29.
Forx fixed, the random field (Yr,t(x),0≤r≤t≤T) admits a continuous version. Moreover,
(70)E[|Yr,s(x)-Yr′,s′(x)|p]≤c(1+|x|p)(|s′-s|pη+|r-r′|pη).
We still denote by Y this continuous version.Proof.
Without loss of generality, assume thats≤s′ and remark that Ys,s′(x) thus belongs to σ{B˘uT,u≥s}:
(71)E[|Yr,s(x)-Yr′,s′(x)|p]≤c(E[|Yr,s(x)-Yr′,s(x)|p]+E[|Yr′,s(x)-Yr′,s′(x)|p])=c(E[|Yr,s(x)-Yr′,s(x)|p]+E[|Yr′,s(x)-Yr′,s(Ys,s′(x))|p])=c(E[|Zs-r,s(x)-Zs-r′,s(x)|p]+E[|Zs-r′,s(x)-Zs-r′,s(Ys,s′(x))|p]).
According to Theorem 37,
(72)E[|Zs-r,s(x)-Zs-r′,s(x)|p]≤c|r-r′|pη(1+|x|p).
In view of Theorem 21, the stochastic integral which appears in (C) is also a Stratonovitch integral; hence, we can apply the substitution formula and say
(73)Zs-r′,s(Ys,s′(x))=Zs-r′,s(y)|y=Ys,s′(x).
Thus we can apply Theorem 37 and obtain that
(74)E[|Zs-r′,s(x)-Zs-r′,s(Ys,s′(x))|p]≤cE[|x-Ys,s′(x)|p].
The right hand side of this equation is in turn equal to E[|Z0,s′-Zs′-s,s′(x)|p] thus, we get
(75)E[|Zs-r′,s(x)-Zs-r′,s(Ys,s′(x))|p]≤c(1+|x|p)|s′-s|pη.
Combining (72) and (75) gives
(76)E[|Yr,s(x)-Yr′,s′(x)|p]≤c(1+|x|p)(|s′-s|pη+|r-r′|pη),
hence the result comesThus, we have the main result of this paper.Theorem 30.
Assume thatV˘T is an E0-causal map continuous from ℒp into ℐα,p for α>0 and p≥4 such that αp>1. Then (A) has one and only one solution.Proof.
Under the hypothesis, we know that(B) has a unique solution which satisfies (67). By definition a solution of (B), the process Y-1:(ω,s)↦Yst-1(ω,x) belongs to 𝕃p,1; hence, we can apply the substitution formula. Following the lines of proof of the previous theorem, we see that Y-1 is a solution of (A).
In the reverse direction, two distinct solutions of(A) would give rise to two solutions of (B) by the same principles. Since this is definitely impossible in view of Corollary 28  (A) has at most one solution.


## 5. Technical Proofs

### 5.1. Substitution Formula

The proof of Theorem21 relies on several lemmas including one known in anticipative calculus as the substitution formula, compare [38].Theorem 31.
Assume that Hypothesis1 (p,η) holds. Let u belong to 𝕃p,1. If V∇Wu is of trace class, then
(77)∫0T‍DWu(s)ds=trace(V∇Wu).
Moreover,
(78)E[|trace(V∇Wu)|p]≤c∥u∥𝕃p,1p.Proof.
For eachk, let (ϕk,m,m=1,…,2k) be the functions ϕk,m=2k/21[(m-1)2-k,m2-k). Let Pk be the projection onto the span of the ϕk,m; since ∇WVu is of trace class, we have (see [42])
(79)trace(V∇Wptu)=limk→+∞trace(PkV∇WptuPk).
Now,
(80)trace(PkV∇WuPk)=∑m=1k‍(V∇Wptu,ϕk,m⊗ϕk,m)ℒ2⊗ℒ2=∑m=1k‍2k∫(m-1)2-k∧tm2-k∧t‍∫(m-1)2-k∧tm2-k∧t‍V(∇rWu)(s)dsdr.
According to the proof of Theorem 20, the first part of the theorem follows. The second part is then a rewriting of (51).Forp≥1, let Γp be the set of random fields:
(81)u:Rm→𝕃p,1x↦((ω,s)↦u(ω,s,x))
equipped with the seminorms,
(82)pK(u)=supx∈K∥u(x)∥𝕃p,1
for any compact K of Rm.Corollary 32 (substitution formula).
Assume that Hypothesis1 (p,η) holds. Let {u(x),x∈Rm} belong to Γp. Let F be a random variable such that ((ω,s)↦u(ω,s,F)) belongs to 𝕃p,1. Then,
(83)∫0T‍u(s,F)∘dWV(s)=∫0T‍u(s,x)∘dWsV|x=F.Proof.
Simple random fields of the form:(84)u(ω,s,x)=∑l=1K‍Hl(x)ul(ω,s)
with Hl smooth and ul in 𝕃p,1 are dense in Γp. In view of (53), it is sufficient to prove the result for such random fields. By linearity, we can reduce the proof to random fields of the form H(x)u(ω,s). Now for any partition π,
(85)δW(∑ti∈π‍1θi∫ti∧tti+1∧t‍H(F)V(u(ω,.))(r)dr1[ti,ti+1))=H(F)δW(∑ti∈π‍1θi∫ti∧tti+1∧t‍V(u(ω,.))(r)dr1[ti,ti+1))-∑ti∈π‍∫ti∧tti+1∧t‍∫ti∧tti+1∧t‍H′(F)∇sWFVu(r)dsdr.
On the other hand,
(86)∇sW(H(F)u(ω,r))=H′(F)∇sWFu(r),
Hence,
(87)∑ti∈π‍1θi∬[ti∧t,ti+1∧t]2V(∇rWH(F)u)(s)dsdr=∑ti∈π‍1θi∬[ti∧t,ti+1∧t]2H′(F)∇sWFVu(r)dsdr.
According to Theorem 20, (83) is satisfied for simple random fields.Definition 33.
For any0≤r≤t≤T, for u in 𝕃p,1, we define ∫rt‍u(s)∘dWV(s) as
(88)∫rt‍u(s)∘dWV(s)=∫0t‍u(s)∘dWV(s)-∫0r‍u(s)∘dWV(s)=∫0T‍etu(s)dWV(s)-∫0T‍eru(s)∘dWV(s)=δW(V(et-er)u)+∫rt‍DWu(s)ds.By the very definition of trace class operators, the next lemma is straightforward.Lemma 34.
LetA and B be two continuous maps from ℒ2([0,T];Rn) into itself. Then, the map τTA⊗B (resp. AτT⊗B) is of trace class if and only if the map A⊗τTB (resp. A⊗BτT) is of trace class. Moreover, in such a situation,
(89)trace(τTA⊗B)=trace(A⊗τTB),resp.trace(AτT⊗B)=trace(A⊗BτT).The next corollary follows by a classical density argument.Corollary 35.
Letu∈𝕃2,1 such that ∇W⊗τTVu and ∇W⊗VτTu are of trace class. Then, τT∇W⊗Vu and ∇WτT⊗Vu are of trace class. Moreover, we have
(90)trace(∇W⊗τTVu)=trace(τT∇W⊗Vu),trace(∇W⊗(VτT)u)=trace(∇WτT⊗Vu).Proof of Theorem21.
We first study the divergence term. In view of Theorem3 we have
(91)δB(V(eT-r-eT-t)τTu˘∘ΘT)=δB(VτT(et-er)u˘∘ΘT)=δB(τTV˘T(et-er)u˘∘ΘT)=δ˘(V˘T(et-er)u˘)(ω˘)=∫rt‍V˘T(1[r,t]u˘)(s)dBT(s).
According to Lemma 15  (V˘T)* is E˘0 causal, and, according to Lemma 9, it is strictly E˘0 causal. Thus, Theorem 14 implies that ∇˘V(et-er)u˘ is of trace class and quasinilpotent. Hence Corollary 35 induces that
(92)τTV˘TτT⊗τT∇˘τT(et-er)u˘
is trace class and quasinilpotent. Now, according to Theorem 2, we have
(93)τTV˘TτT⊗τT∇˘τT(et-er)u˘=V(∇τT(eT-r-eT-t)u˘∘ΘT).
According to Theorem 20, we have proved (54).


### 5.2. The Forward Equation

Lemma 36.
Assume that Hypothesis1(p,η) holds and that σ is Lipschitz continuous. Then, for any 0≤a≤b≤T, the map
(94)V˘T∘σ:C([0,T],Rn)→C([0,T],Rn)ϕ↦V˘T(σ∘ψ1[a,b])
is Lipschitz continuous and Gâteaux differentiable. Its differential is given by
(95)dV˘T∘σ(ϕ)[ψ]=V˘T(σ′∘ϕψ).
Assume furthermore that σ is sublinear; that is,
(96)|σ(x)|≤c(1+|x|),foranyx∈Rn.
Then, for any ψ∈C([0,T],Rn), for any t∈[0,T],
(97)|V˘T(σ∘ψ)(t)|≤cTη+1/p(1+∫0t‍|ψ(s)|pds)≤cTη+1/p(1+∥ψ∥∞).Proof.
Letψ and ϕ be two continuous functions, since C([0,T],Rn) is continuously embedded in ℒp and V˘T(σ∘ψ-σ∘ϕ) belongs to Hol(η). Moreover,
(98)supt≤T|V˘T(σ∘ψ1[a,b])(t)-V˘T(σ∘ϕ1[a,b])(t)|≤c∥V˘T((σ∘ψ-σ∘ϕ)1[a,b])∥Hol(η)≤c∥(σ∘ψ-σ∘ϕ)1[a,b]∥ℒp≤c∥ϕ-ψ∥ℒp([a,b])≤csupt≤T|ψ(t)-ϕ(t)|,
since σ is Lipschitz continuous.
Letϕ and ψ be two continuous functions on [0,T]. Since σ is Lipschitz continuous, we have
(99)σ(ψ(t)+εϕ(t))=σ(ψ(t))+ε∫01‍σ′(uψ(t)+(1-u)ϕ(t))du.
Moreover, since σ is Lipschitz, σ′ is bounded and
(100)∫0T‍|∫01‍σ′(uψ(t)+(1-u)ϕ(t))du|pdt≤cT.
This means that (t↦∫01‍σ′(uψ(t)+(1-u)ϕ(t))du) belongs to ℒp. Hence, according to Hypothesis 1,
(101)∥V˘T(∫01‍σ′(uψ(.)+(1-u)ϕ(.))du)∥C≤cT.
Thus,
(102)limε→0ε-1(V˘T(σ∘(ψ+εϕ))-V˘T(σ∘ψ))exists,
and V˘T∘σ is Gâteaux differentiable and its differential is given by (95).
Sinceσ∘ψ belongs to C([0,T],Rn), according to Hypothesis 1, we have
(103)|V˘T(σ∘ψ)(t)|≤c(∫0t‍sηp|σ(ψ(s))|pds)1/p≤cTη(∫0t‍(1+|ψ(s)|p)ds)1/p≤cTη+1/p(1+∥ψ∥∞p)1/p≤cTη+1/p(1+∥ψ∥∞).
The proof is thus complete.Following [43], we then have the following nontrivial result.Theorem 37.
Assume that Hypothesis1  (p,η) holds and that σ is Lipschitz continuous. Then, there exists one and only one measurable map from Ω×[0,T]×[0,T] into 𝔊 which satisfies the first two points of Definition (C). Moreover,
(104)E[|Zr,t(x)-Zr′,t(x′)|p]≤c(1+|x|p∨|x′|p)×(|r-r′|pη+|x-x′|p),
and, for any x∈Rn, for any 0≤r≤t≤T, we have
(105)E[|Zr,t(x)|p]≤c(1+|x|p)ecTηp+1.
Note even if x and x′ are replaced by σ{B˘T(u),t≤u} measurable random variables, the last estimates still hold.Proof.
Existence, uniqueness, and homeomorphy of a solution of(C) follow from [43]. The regularity with respect to r and x is obtained as usual by BDG inequality and Gronwall Lemma. For x or x′ random, use the independence of σ{B˘T(u),t≤u} and σ{B˘T(u),r∧r′≤u≤t}.Theorem 38.
Assume that Hypothesis1  (p,η) holds and that σ is Lipschitz continuous and sublinear. Then, for any x∈Rn, for any 0≤r≤s≤t≤T, (ω,r)↦Zr,s(ω,Zs,t(x)) and (ω,r)↦Zr,t-1(ω,x) belong to 𝕃p,1.Proof.
According to ([44], Theorem 3.1), the differentiability of ω↦Zr,t(ω,x) is ensured. Furthermore,
(106)∇uZr,t=-V˘T(σ∘Z.,t1[r,t])(u)-∫rt‍V˘T(σ′(Z.,t).∇uZ.,t1[r,t])(s)dB˘(s),
where σ′ is the differential of σ. For M>0, let
(107)ξM=inf{τ,|∇uZτ,t|p≥M},Zτ,tM=Zτ∨ξM,t.
Since V˘T is continuous from ℒp into itself and σ is Lipschitz, according to BDG inequality, for r≤u,
(108)E[|∇uZr,tM|p]≤cE[|V˘T(σ∘Z.,tM1[r,t])(u)|p]+cE[∫rt‍|V˘T(σ′(Z.,tM)∇uZ.,tM1[r,t])(s)|pds]≤c(1+E[∫rt‍upη∫ru‍|Zτ,t|pdτdu]+E[∫rt‍spη∫rs‍|∇uZτ,tM|pdτds])≤c(1+E[∫rt‍|Zτ,t|p(tpη+1-τpη+1)dτ]+E[∫rt‍|∇uZτ,tM|p(tpη+1-τpη+1)dτ])≤ctpη+1(1+E[∫rt‍|Zτ,t|pdτ]+E[∫rt‍|∇uZτ,tM|pdτ]).
Then, Gronwall Lemma entails that
(109)E[|∇uZr,tM|p]≤c(1+E[∫rt‍|Zτ,t|pdτ]),
hence by Fatou lemma,
(110)E[|∇uZr,t|p]≤c(1+E[∫rt‍|Zτ,t|pdτ]).
The integrability of E[|∇uZr,t|p] with respect to u follows.
Now, since0≤r≤s≤t≤T, Zs,t(x) is independent of Zr,s(x), thus the previous computations still hold and (ω,r)↦Zr,s(ω,Zs,t(x)) belongs to 𝕃p,1.
According to [45], to prove that Zr,t-1(x) belongs to 𝔻p,1, we need to prove that (1)
for everyh∈ℒ2, there exists an absolutely continuous version of the process (t↦Zr,t-1(ω+th,x)),(2)
there existsDZr,t-1, an ℒ2-valued random variable such that, for every h∈ℒ2,
(111)1t(Zr,t-1(ω+th,x)-Zr,t-1(ω,x))→t→0∫0T‍DZr,t-1(s)h(s)ds,
where the convergence holds in probability,(3)
DZr,t-1 belongs to ℒ2(Ω,ℒ2). We first show that
(112)E[|∂Zr,t∂x(ω,Zr,t-1(x))|-p]isfinite.
Since
(113)∂Zr,t∂x(ω,x)=Id+∫rt‍V˘T(σ′(Z.,t(x))∂Z.,t(ω,x)∂x)(s)dB˘(s),
let Θv=supu≤v|∂xZu,t(x)|. The same kind of computations as above entails that (for the sake of brevity, we do not detail the localisation procedure as it is similar to the previous one)
(114)E[Θv2q]≤c+cE[∫ut‍Θs2(q-1)(∫us‍|∂xZτ,t(x)|p∣dτ)2/pds]+cE[(∫ut‍Θsq-2(∫us‍|∂xZτ,t(x)|p∣dτ)2/p)2ds].
Hence,
(115)E[Θv2q]≤c(1+∫vt‍E[Θs2q]ds),
and (112) follows by Fatou and Gronwall lemmas. Since Zr,t(ω,Zr,t-1(ω,x))=x, the implicit function theorem implies that Zr,t-1(x) satisfies the first two properties and that
(116)∇Zr,t(ω,Zr,t-1(x))+∂Zr,t∂x(ω,Zr,t-1(x))∇~Zr,t-1(ω,x).
It follows by Hölder inequality and (112) that
(117)∥DZr,t-1(x))∥p,1≤c∥Zr,t(x))∥2p,1∥(∂xZr,t(x))-1∥2p;
hence Zr,t-1 belongs to 𝕃p,1.


## 5.1. Substitution Formula

The proof of Theorem21 relies on several lemmas including one known in anticipative calculus as the substitution formula, compare [38].Theorem 31.
Assume that Hypothesis1 (p,η) holds. Let u belong to 𝕃p,1. If V∇Wu is of trace class, then
(77)∫0T‍DWu(s)ds=trace(V∇Wu).
Moreover,
(78)E[|trace(V∇Wu)|p]≤c∥u∥𝕃p,1p.Proof.
For eachk, let (ϕk,m,m=1,…,2k) be the functions ϕk,m=2k/21[(m-1)2-k,m2-k). Let Pk be the projection onto the span of the ϕk,m; since ∇WVu is of trace class, we have (see [42])
(79)trace(V∇Wptu)=limk→+∞trace(PkV∇WptuPk).
Now,
(80)trace(PkV∇WuPk)=∑m=1k‍(V∇Wptu,ϕk,m⊗ϕk,m)ℒ2⊗ℒ2=∑m=1k‍2k∫(m-1)2-k∧tm2-k∧t‍∫(m-1)2-k∧tm2-k∧t‍V(∇rWu)(s)dsdr.
According to the proof of Theorem 20, the first part of the theorem follows. The second part is then a rewriting of (51).Forp≥1, let Γp be the set of random fields:
(81)u:Rm→𝕃p,1x↦((ω,s)↦u(ω,s,x))
equipped with the seminorms,
(82)pK(u)=supx∈K∥u(x)∥𝕃p,1
for any compact K of Rm.Corollary 32 (substitution formula).
Assume that Hypothesis1 (p,η) holds. Let {u(x),x∈Rm} belong to Γp. Let F be a random variable such that ((ω,s)↦u(ω,s,F)) belongs to 𝕃p,1. Then,
(83)∫0T‍u(s,F)∘dWV(s)=∫0T‍u(s,x)∘dWsV|x=F.Proof.
Simple random fields of the form:(84)u(ω,s,x)=∑l=1K‍Hl(x)ul(ω,s)
with Hl smooth and ul in 𝕃p,1 are dense in Γp. In view of (53), it is sufficient to prove the result for such random fields. By linearity, we can reduce the proof to random fields of the form H(x)u(ω,s). Now for any partition π,
(85)δW(∑ti∈π‍1θi∫ti∧tti+1∧t‍H(F)V(u(ω,.))(r)dr1[ti,ti+1))=H(F)δW(∑ti∈π‍1θi∫ti∧tti+1∧t‍V(u(ω,.))(r)dr1[ti,ti+1))-∑ti∈π‍∫ti∧tti+1∧t‍∫ti∧tti+1∧t‍H′(F)∇sWFVu(r)dsdr.
On the other hand,
(86)∇sW(H(F)u(ω,r))=H′(F)∇sWFu(r),
Hence,
(87)∑ti∈π‍1θi∬[ti∧t,ti+1∧t]2V(∇rWH(F)u)(s)dsdr=∑ti∈π‍1θi∬[ti∧t,ti+1∧t]2H′(F)∇sWFVu(r)dsdr.
According to Theorem 20, (83) is satisfied for simple random fields.Definition 33.
For any0≤r≤t≤T, for u in 𝕃p,1, we define ∫rt‍u(s)∘dWV(s) as
(88)∫rt‍u(s)∘dWV(s)=∫0t‍u(s)∘dWV(s)-∫0r‍u(s)∘dWV(s)=∫0T‍etu(s)dWV(s)-∫0T‍eru(s)∘dWV(s)=δW(V(et-er)u)+∫rt‍DWu(s)ds.By the very definition of trace class operators, the next lemma is straightforward.Lemma 34.
LetA and B be two continuous maps from ℒ2([0,T];Rn) into itself. Then, the map τTA⊗B (resp. AτT⊗B) is of trace class if and only if the map A⊗τTB (resp. A⊗BτT) is of trace class. Moreover, in such a situation,
(89)trace(τTA⊗B)=trace(A⊗τTB),resp.trace(AτT⊗B)=trace(A⊗BτT).The next corollary follows by a classical density argument.Corollary 35.
Letu∈𝕃2,1 such that ∇W⊗τTVu and ∇W⊗VτTu are of trace class. Then, τT∇W⊗Vu and ∇WτT⊗Vu are of trace class. Moreover, we have
(90)trace(∇W⊗τTVu)=trace(τT∇W⊗Vu),trace(∇W⊗(VτT)u)=trace(∇WτT⊗Vu).Proof of Theorem21.
We first study the divergence term. In view of Theorem3 we have
(91)δB(V(eT-r-eT-t)τTu˘∘ΘT)=δB(VτT(et-er)u˘∘ΘT)=δB(τTV˘T(et-er)u˘∘ΘT)=δ˘(V˘T(et-er)u˘)(ω˘)=∫rt‍V˘T(1[r,t]u˘)(s)dBT(s).
According to Lemma 15  (V˘T)* is E˘0 causal, and, according to Lemma 9, it is strictly E˘0 causal. Thus, Theorem 14 implies that ∇˘V(et-er)u˘ is of trace class and quasinilpotent. Hence Corollary 35 induces that
(92)τTV˘TτT⊗τT∇˘τT(et-er)u˘
is trace class and quasinilpotent. Now, according to Theorem 2, we have
(93)τTV˘TτT⊗τT∇˘τT(et-er)u˘=V(∇τT(eT-r-eT-t)u˘∘ΘT).
According to Theorem 20, we have proved (54).


## 5.2. The Forward Equation

Lemma 36.
Assume that Hypothesis1(p,η) holds and that σ is Lipschitz continuous. Then, for any 0≤a≤b≤T, the map
(94)V˘T∘σ:C([0,T],Rn)→C([0,T],Rn)ϕ↦V˘T(σ∘ψ1[a,b])
is Lipschitz continuous and Gâteaux differentiable. Its differential is given by
(95)dV˘T∘σ(ϕ)[ψ]=V˘T(σ′∘ϕψ).
Assume furthermore that σ is sublinear; that is,
(96)|σ(x)|≤c(1+|x|),foranyx∈Rn.
Then, for any ψ∈C([0,T],Rn), for any t∈[0,T],
(97)|V˘T(σ∘ψ)(t)|≤cTη+1/p(1+∫0t‍|ψ(s)|pds)≤cTη+1/p(1+∥ψ∥∞).Proof.
Letψ and ϕ be two continuous functions, since C([0,T],Rn) is continuously embedded in ℒp and V˘T(σ∘ψ-σ∘ϕ) belongs to Hol(η). Moreover,
(98)supt≤T|V˘T(σ∘ψ1[a,b])(t)-V˘T(σ∘ϕ1[a,b])(t)|≤c∥V˘T((σ∘ψ-σ∘ϕ)1[a,b])∥Hol(η)≤c∥(σ∘ψ-σ∘ϕ)1[a,b]∥ℒp≤c∥ϕ-ψ∥ℒp([a,b])≤csupt≤T|ψ(t)-ϕ(t)|,
since σ is Lipschitz continuous.
Letϕ and ψ be two continuous functions on [0,T]. Since σ is Lipschitz continuous, we have
(99)σ(ψ(t)+εϕ(t))=σ(ψ(t))+ε∫01‍σ′(uψ(t)+(1-u)ϕ(t))du.
Moreover, since σ is Lipschitz, σ′ is bounded and
(100)∫0T‍|∫01‍σ′(uψ(t)+(1-u)ϕ(t))du|pdt≤cT.
This means that (t↦∫01‍σ′(uψ(t)+(1-u)ϕ(t))du) belongs to ℒp. Hence, according to Hypothesis 1,
(101)∥V˘T(∫01‍σ′(uψ(.)+(1-u)ϕ(.))du)∥C≤cT.
Thus,
(102)limε→0ε-1(V˘T(σ∘(ψ+εϕ))-V˘T(σ∘ψ))exists,
and V˘T∘σ is Gâteaux differentiable and its differential is given by (95).
Sinceσ∘ψ belongs to C([0,T],Rn), according to Hypothesis 1, we have
(103)|V˘T(σ∘ψ)(t)|≤c(∫0t‍sηp|σ(ψ(s))|pds)1/p≤cTη(∫0t‍(1+|ψ(s)|p)ds)1/p≤cTη+1/p(1+∥ψ∥∞p)1/p≤cTη+1/p(1+∥ψ∥∞).
The proof is thus complete.Following [43], we then have the following nontrivial result.Theorem 37.
Assume that Hypothesis1  (p,η) holds and that σ is Lipschitz continuous. Then, there exists one and only one measurable map from Ω×[0,T]×[0,T] into 𝔊 which satisfies the first two points of Definition (C). Moreover,
(104)E[|Zr,t(x)-Zr′,t(x′)|p]≤c(1+|x|p∨|x′|p)×(|r-r′|pη+|x-x′|p),
and, for any x∈Rn, for any 0≤r≤t≤T, we have
(105)E[|Zr,t(x)|p]≤c(1+|x|p)ecTηp+1.
Note even if x and x′ are replaced by σ{B˘T(u),t≤u} measurable random variables, the last estimates still hold.Proof.
Existence, uniqueness, and homeomorphy of a solution of(C) follow from [43]. The regularity with respect to r and x is obtained as usual by BDG inequality and Gronwall Lemma. For x or x′ random, use the independence of σ{B˘T(u),t≤u} and σ{B˘T(u),r∧r′≤u≤t}.Theorem 38.
Assume that Hypothesis1  (p,η) holds and that σ is Lipschitz continuous and sublinear. Then, for any x∈Rn, for any 0≤r≤s≤t≤T, (ω,r)↦Zr,s(ω,Zs,t(x)) and (ω,r)↦Zr,t-1(ω,x) belong to 𝕃p,1.Proof.
According to ([44], Theorem 3.1), the differentiability of ω↦Zr,t(ω,x) is ensured. Furthermore,
(106)∇uZr,t=-V˘T(σ∘Z.,t1[r,t])(u)-∫rt‍V˘T(σ′(Z.,t).∇uZ.,t1[r,t])(s)dB˘(s),
where σ′ is the differential of σ. For M>0, let
(107)ξM=inf{τ,|∇uZτ,t|p≥M},Zτ,tM=Zτ∨ξM,t.
Since V˘T is continuous from ℒp into itself and σ is Lipschitz, according to BDG inequality, for r≤u,
(108)E[|∇uZr,tM|p]≤cE[|V˘T(σ∘Z.,tM1[r,t])(u)|p]+cE[∫rt‍|V˘T(σ′(Z.,tM)∇uZ.,tM1[r,t])(s)|pds]≤c(1+E[∫rt‍upη∫ru‍|Zτ,t|pdτdu]+E[∫rt‍spη∫rs‍|∇uZτ,tM|pdτds])≤c(1+E[∫rt‍|Zτ,t|p(tpη+1-τpη+1)dτ]+E[∫rt‍|∇uZτ,tM|p(tpη+1-τpη+1)dτ])≤ctpη+1(1+E[∫rt‍|Zτ,t|pdτ]+E[∫rt‍|∇uZτ,tM|pdτ]).
Then, Gronwall Lemma entails that
(109)E[|∇uZr,tM|p]≤c(1+E[∫rt‍|Zτ,t|pdτ]),
hence by Fatou lemma,
(110)E[|∇uZr,t|p]≤c(1+E[∫rt‍|Zτ,t|pdτ]).
The integrability of E[|∇uZr,t|p] with respect to u follows.
Now, since0≤r≤s≤t≤T, Zs,t(x) is independent of Zr,s(x), thus the previous computations still hold and (ω,r)↦Zr,s(ω,Zs,t(x)) belongs to 𝕃p,1.
According to [45], to prove that Zr,t-1(x) belongs to 𝔻p,1, we need to prove that (1)
for everyh∈ℒ2, there exists an absolutely continuous version of the process (t↦Zr,t-1(ω+th,x)),(2)
there existsDZr,t-1, an ℒ2-valued random variable such that, for every h∈ℒ2,
(111)1t(Zr,t-1(ω+th,x)-Zr,t-1(ω,x))→t→0∫0T‍DZr,t-1(s)h(s)ds,
where the convergence holds in probability,(3)
DZr,t-1 belongs to ℒ2(Ω,ℒ2). We first show that
(112)E[|∂Zr,t∂x(ω,Zr,t-1(x))|-p]isfinite.
Since
(113)∂Zr,t∂x(ω,x)=Id+∫rt‍V˘T(σ′(Z.,t(x))∂Z.,t(ω,x)∂x)(s)dB˘(s),
let Θv=supu≤v|∂xZu,t(x)|. The same kind of computations as above entails that (for the sake of brevity, we do not detail the localisation procedure as it is similar to the previous one)
(114)E[Θv2q]≤c+cE[∫ut‍Θs2(q-1)(∫us‍|∂xZτ,t(x)|p∣dτ)2/pds]+cE[(∫ut‍Θsq-2(∫us‍|∂xZτ,t(x)|p∣dτ)2/p)2ds].
Hence,
(115)E[Θv2q]≤c(1+∫vt‍E[Θs2q]ds),
and (112) follows by Fatou and Gronwall lemmas. Since Zr,t(ω,Zr,t-1(ω,x))=x, the implicit function theorem implies that Zr,t-1(x) satisfies the first two properties and that
(116)∇Zr,t(ω,Zr,t-1(x))+∂Zr,t∂x(ω,Zr,t-1(x))∇~Zr,t-1(ω,x).
It follows by Hölder inequality and (112) that
(117)∥DZr,t-1(x))∥p,1≤c∥Zr,t(x))∥2p,1∥(∂xZr,t(x))-1∥2p;
hence Zr,t-1 belongs to 𝕃p,1.



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