# A Stability Result for Stochastic Differential Equations Driven by Fractional Brownian Motions

Authors: Bruno Saussereau
Journal: International Journal of Stochastic Analysis
 (2012)

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## Abstract

We study the stability of the solutions of stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than  half. We prove that when the initial conditions, the drift, and the diffusion coefficients as well as the fractional Brownian motions converge in a suitable sense, then the sequence of the solutions of the corresponding equations converge in Hölder norm to the solution of a stochastic differential equation. The limit equation is driven by the limit fractional Brownian motion and its coefficients are the limits of the sequence of the coefficients.


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## Body

## 1. Introduction and Main Result

Suppose thatBH=(BtH)0≤t≤T is an m-dimensional fractional Brownian motion (fBm in short) with Hurst parameter H defined on a complete filtered probability space (Ω,ℱ,(ℱt)0≤t≤T,P). We mean that the components BH,j, j=1,…,m are independent centered Gaussian processes with the covariance function
(1.1)RH(s,t)=12(t2H+s2H-|t-s|2H).
If H=1/2, then BH is clearly a Brownian motion. Since for any p≥1, E|BtH,j-BsH,j|p=cp|t-s|pH the processes BH,j have α-Hölder continuous paths for all α∈(0,H) (see [1] for further information about fBm).In this paper we fix1/2<H<1 and we consider the solution (Xt)0≤t≤T of the following stochastic differential equation (abbreviated by SDE from now on) on Rd(1.2)Xti=xi+∑j=1m‍∫0t‍σi,j(Xs)dBsH,j+∫0t‍bi(Xs)ds,0≤t≤T,i=1,…,d, x∈Rd is the initial value of the process X.Under suitable assumptions onσ, the processes σ(X) and BH have trajectories which are Hölder continuous of order strictly larger than 1/2 so we can use the integral introduced by Young in [2]. The stochastic integral in (1.2) is then a path-wise Riemann-Stieltjes integral. A first result on the existence and uniqueness of a solution of such an equation was obtained in [3] using the notion of p-variation. The theory of rough paths introduced by Lyons in [3] was used by Coutin and Qian in order to prove an existence and uniqueness result for (1.2) (see [4]). The Riemann-Stieltjes integral appearing in (1.2) can be expressed as a Lebesgue integral using a fractional integration by parts formula (see Zähle [5]). Using this formula Nualart and Răşcanu have established in [6] the existence of a unique solution for a class of general differential equations that includes (1.2). Later on, the regularity in the sense of Malliavin calculus and the absolute continuity of the law of the random variables Xt have been investigated in [7–10].In order to obtain moment bounds on the solution of (1.2), we have to estimate the corresponding determinist differential equation very carefully. Indeed, an exponential of the Hölder norm of the fBm may appear and by Fernique’s theorem, it is well known that such exponential moment does not always exist. This fact will be specified in Section 2. Thanks to a technical trick due to Hairer and Pillai in [11] (see also [12]), some estimations that are compatible with exponential moments are now available. This is the starting point of this short communication: first we will estimate the difference between two solutions of SDEs with different coefficients. We will endeavor ourselves to give some bounds that are suitable for stability results.Now we present the kind of results we are interested in and so we need further notations. For a differentiable functionφ from Rd to Rq, we denote (if the following quantities do exist) ∥φ∥∞=maxj=1,…,q∥φj∥∞ and
(1.3)∥φ∥C1=∥φ∥∞+maxi=1,…,dmaxj=1,…,q∥∂φj∂xi∥∞.
The space C1 is the space of continuously differentiable functions φ such that ∥φ∥C1<∞. The space C2 is defined in a similar way. For 0<λ<1 and 0≤a<b≤T, we denote by Cλ(a,b;Rd) the space of λ-Hölder continuous functions f:[a,b]→Rd, equipped with the norm
(1.4)∥f∥λ:=∥f∥a,b,∞+∥f∥a,b,λ,
where
(1.5)∥f∥a,b,∞=supa≤r≤b|f(r)|,∥f∥a,b,λ=supa≤r≤s≤b|f(s)-f(r)||s-r|λ.
We simply write Cλ(a,b) when d=1.The main result of this work is the following theorem.Theorem 1.1.
Let(BH,n)n≥1 be a sequence of fractional Brownian motions defined on (Ω,ℱ,(ℱt)0≤t≤T,P). Let (xn)n≥1, (bn)n≥1, and (σn)n≥1 be some sequences in Rd, C1, and C2. One considers the sequence (Xn)n≥1 of processes such that for any n≥1, (Xtn)0≤t≤T is the unique solution of
(1.6)Xtn=xn+∫0t‍σn(Xsn)dBsH,n+∫0t‍bn(Xsn)ds,0≤t≤T.
If there exists(i)
x∈Rd, b∈C1 and σ∈C2 such that
(1.7)limn→∞{|xn-x|+∥bn-b∥∞+∥σ-σn∥C1}=0;(ii)
a fractional Brownian motionBH defined on (Ω,ℱ,(ℱt)0≤t≤T,P) such that for β∈(1/2,H) and p≥1 one has
(1.8)limn→∞E(∥BH,n-BH∥0,T,βp)=0;then
(1.9)limn→∞E(∥Xn-X∥βp)=0,
where (Xt)0≤t≤T is the solution of (1.2) with the coefficients x, b, and σ and driven by the fBm BH.As usual in the theory of SDEs driven by fBm, the above theorem will be the counterpart of a deterministic result on ordinary differential equations driven by Hölder continuous functions. More precisely, Theorem1.1 will be a consequence of an estimation on the Hölder norm of the difference of two solutions of rough differential equations. This result is interesting in itself and it is the subject of Section 2. It is precisely stated in Proposition 2.3 but we present here a brief description of the result we have obtained. We consider for some β∈(1/2,1) two deterministic rough functions g and g~ in Cβ and two deterministic differential equations
(1.10)xt=x0+∫0t‍b(xs)ds+∫0t‍σ(xs)dgs,x~t=x~0+∫0t‍b~(x~s)ds+∫0t‍σ~(x~s)dg~s,
where all the coefficients are smooth. Then we will prove that there exists a constant C such that
(1.11)∥x-x~∥β≤C(|x0-x~0|+∥b-b~∥∞+∥σ-σ~∥C1∥g∥0,T,β+∥g-g~∥0,T,β)×(1+∥g∥0,T,β)2/βexp{C∥g∥0,T,β1/β}.
Since our upper bound is explicit, this estimate can be viewed as a refinement of Theorems 11.3 and 11.6 of [13]. Nevertheless we strength the fact that many results have been obtained in the theory of fractional SDEs thanks to rough paths theory. We may prove the above results by rough paths techniques but we adopt the simplest context of Young’s integral. The stability with respect to the driving noise is a reformulation of the continuity of the Itô map and this is well known. A weaker stability result with respect to the initial condition is proved in [14]. The stability with respect to all the coefficients in (1.2) is new to our knowledge.The paper is organized as follows. In Section2 we present the case of deterministic differential equations driven by Hölder continuous function and we state an estimation on the difference between the solutions of such equations (see Proposition 2.3). Theorem 1.1 will be a straightforward consequence of this work on deterministic differential equations. Finally some proofs are gathered in Section 3 and in the appendix.


## 2. Deterministic Differential Equation Driven by Rough Functions

This section deals with deterministic differential equations driven by Hölder’s continuous functions. These equations are the one satisfied by the trajectories of the solution of (1.2). Our aim is to prove an estimate for the difference of two solutions of deterministic differential equations driven by two different Hölder continuous functions. In [8, Theorem 3.3], such estimates are proved but are unfortunately unusable in our context (see the discussion below). Proposition 2.3 hereafter will strengthen the result of [8].Suppose thatf∈Cλ(a,b) and g∈Cμ(a,b) with λ+μ>1. From [2], the Riemann-Stieltjes integral ∫ab‍fdg exists. In [5], the author provides an explicit expression for the integral ∫ab‍fdg in terms of fractional derivatives. In order to give some precisions, we consider α>0 being such that β>1-α. Supposing that the following limit exists and is finite, we denote gb-(t)=g(t)-limε↓0g(b-ε). Then the Riemann-Stieltjes integral can be expressed as
(2.1)∫ab‍ftdgt=(-1)α∫ab‍(Da+αf)(t)(Db-1-αgb-)(t)dt,
where
(2.2)Da+αf(t)=1Γ(1-α)(f(t)(t-a)α+α∫at‍f(t)-f(s)(t-s)α+1ds),Db-αgb-(t)=(-1)αΓ(1-α)(g(t)-g(b)(b-t)α+α∫tb‍g(t)-g(s)(s-t)α+1ds).
We refer to [15] for further details on fractional operators. The following useful lemma is now classical. Its proof is postponed in the appendix.Lemma 2.1.
Letf and g in Cβ(s,t) with 1/2<β<1 and 0≤s<t≤T, then there exists a universal constant κ such that
(2.3)|∫st‍frdgr|≤κβ-1/2∥g∥0,T,β[∥f∥s,t,∞(t-s)β+∥f∥s,t,β(t-s)2β].Set1/2<β<1 and let g,g~∈Cβ(0,T;Rm). We will work with the following deterministic differential equations on Rd:
(2.4)xti=x0i+∫0t‍bi(xs)ds+∑j=1m‍∫0t‍σi,j(xs)dgsj,x~ti=x~0i+∫0t‍b~i(x~s)ds+∑j=1m‍∫0t‍σ~i,j(x~s)dg~sj,
for 0≤t≤T,i=1,…,d and x0,x~0∈Rd.We introduce the following assumptions on the coefficients of the above equations. For a functionφ from Rd to Rq, ∇φ denotes the matrix of first order derivatives and ∇2φ denotes its Hessian.(H1)
There exists some positive constantsb0, b1, b~0, b~1 such that ∥b∥∞≤b0, ∥b~∥∞≤b~0, ∥∇b∥∞≤b1, and ∥∇b~∥∞≤b~1.(H2)
There exists some positive constantsc0, c1, c2, c~0, c~1, and c~2 such that ∥σ∥∞≤c0, ∥∇σ∥∞≤c1, ∥∇2σ∥∞≤c2, ∥σ~∥∞≤c~0, ∥∇σ~∥∞≤c~1, and ∥∇2σ~∥∞≤c~2.It is proved in [6, Theorem 5.1] that if 1-β<α<1/2, each of the above equations has a unique (1-α)-Hölder continuous solution. The estimates on the solution (xt)0≤t≤T obtained in [6] were improved in [8]. Let us recall the following observations concerning Theorem 3 in Hu and Nualart [8] (see also [11] for similar comments). It has been proved in [8] that if b=0 and σ is twice continuously differentiable with bounded second order derivatives, then there exists a constant k that depends on T, β, and σ such that
(2.5)∥x-x~∥0,T,∞≤kexp{k∥g∥0,T,β1/β(1+∥x∥0,T,β+∥x~∥0,T,β)1/β(1+∥x∥0,T,β)∥g-g~∥0,T,βk∥g∥0,T,β1/β(1+∥x∥0,T,β+∥x~∥0,T,β)1/β}.
Replacing g and g~ by the trajectories of the fractional Brownian motions BH and B~H, we obtain estimation in the supremum norm for the difference of the processes X and X~ satisfying
(2.6)Xt=x0+∫0t‍σ(Xs)dBsH,X~t=x0+∫0t‍σ(X~s)dB~sH.
Thus the estimation (2.5) holds almost-surely and one have to take expectation. For this purpose we use the following Fernique’s type result on the exponential moments of the Hölder norm of the fBm.Lemma 2.2.
LetT>0, 1/2<β<H<1. Then for any α<1/(128(2T)2(H-β))(2.7)E[exp(α∥BH∥0,T,β2)]≤(1-128α(2T)2(H-β))-1/2.One refers to [16] for a proof of this lemma. Lemma 2.2 implies the following integrability property for X (or X~): for any λ>0 and γ<2H,
(2.8)E[exp(λ∥X∥0,T,∞γ)]<∞.
Inequality (2.8) together with (2.5) will unfortunately be useless since the quantities ∥X∥0,T,∞1/β and ∥B∥0,T,∞1/β appear in a multiplicative way in the exponent in the right hand side of (2.5). Indeed, Young’s inequality yields ∥X∥0,T,∞1/β∥B∥0,T,∞1/β≤c∥X∥0,T,∞p/β+c∥B∥0,T,∞q/β and if one imposes p/β<2 then necessarily q/β>2 and the finiteness of the expression such as
(2.9)E[exp(k∥B∥0,T,∞1/β∥X∥0,T,∞1/β)]
cannot be deduced from Lemma 2.2 and (2.8) (in fact we do not even know if this expectation is finite). Hence we need a suitable estimate to obtain moment bounds on the quantities we are interested in. Such investigations have been carried out in [11, 12] but we need some nontrivial modifications to handle the difference x-x~ when x and x~ are the solutions of (2.4).Therefore the next result is a strengthening of Theorem  3.3 in [8] and is based on the method used in [11, Lemma 3.2]. It may also be viewed as a refinement of Theorems 11.3 and 11.6 of [13].Proposition 2.3.
LetT be fixed and let g and g~ be Hölder continuous of order 1/2<β<1. Under (H1) and (H2), there exists a constant C that depends only on T, β, b0, b~0, c0, c~0, c1, c~1, c2, and c~2 such that
(2.10)∥x-x~∥β≤C(|x0-x~0|+∥b-b~∥∞+∥σ-σ~∥C1∥g∥0,T,β+∥g-g~∥0,T,β)×(1+∥g∥0,T,β)2/βexp{C∥g∥0,T,β1/β}.It is worth to notice that a careful reading of the proof shows thatC depends continuously on its parameters. This is important when we apply this proposition to stability properties of stochastic differential equations.


## 3. Proofs

The subject of this section is the proof of Proposition2.3 and Theorem 1.1. We follow the arguments developed in the proof of [8, Theorem 3.2], [11, Lemma 3.2] and we give some precisions. We restrict ourselves to the case d=m=1 for simplicity. Thus for a function φ, we denote φ′ its derivative and φ′′ its second order derivative. Moreover cβ:=κ/(β-1/2) designates the constant in (2.3).


### 3.1. A Preliminary Lemma

We will need the following lemma whose proof is borrowed from [11]. For the sake of completeness and to give some information on the constants that are involved in the statement, we briefly recall the arguments that are used in the proof.Lemma 3.1.
There exists an explicit constantM1 that depends on T, b0, b1, c0, c1, and cβ such that
(3.1)∥x∥0,T,∞≤T(1+|x0|)e2M1b1T(2M1(1+∥g∥0,T,β))1/β.Proof.
Let0≤s≤t≤T. Since |b(xr)|≤|b(xr)-b(xs)|+|b(xs)| and |b(xs)|≤b0+b1|xs|, we have
(3.2)|∫st‍b(xr)dr|≤b0(t-s)+b1|xs|(t-s)+b1∥x∥s,t,β(t-s)1+β.
Clearly ∥σ(x·)∥s,t,β≤c1∥x∥s,t,β, thus Inequality (2.3) yields
(3.3)∥x∥s,t,β≤b0(t-s)1-β+b1|xs|(t-s)1-β+b1∥x∥s,t,β(t-s)+cβ∥g∥0,T,β[c0+c1∥x∥s,t,β(t-s)β]≤M1{1+∥g∥0,T,β+b1|xs|(t-s)1-β+(1+∥g∥0,T,β)∥x∥s,t,β(t-s)β}
with M1=max(1,T1-β∥b∥C1,cβ∥σ∥C1). We denote
(3.4)Δ1={2M1(1+∥g∥0,T,β)}-1/β
and when (t-s)≤Δ1 we may write
(3.5)∥x∥s,t,β≤2M1(1+∥g∥0,T,β+b1|xs|Δ11-β).
Since ∥x∥s,t,∞≤|xs|+∥x∥s,t,β(t-s)β, we obtain for (t-s)≤Δ1 that
(3.6)∥x∥s,t,∞≤|xs|(1+2M1b1Δ1)+1.
By induction with N=T/Δ1 it follows that
(3.7)∥x∥0,T,∞≤|x0|(1+2M1b1Δ1)N+∑k=1N‍(1+2M1b1Δ1)k≤(TΔ1)(1+|x0|)(1+2M1b1Δ1)T/Δ1
and since for any a>0, (1+x/a)a≤ex, we have (1+2M1b1Δ1)T/Δ1≤e2M1b1T and we finally deduce (3.1).
As noticed in [11], if b is the null function, then b1=0 and the last above argument fails. In this case it is impossible to obtain a bound without any exponential of the quantity ∥g∥0,T,β.Remark 3.2.
When(t-s)≤Δ1, we substitute (3.1) into (3.5) and we deduce the estimate
(3.8)∥x∥s,t,β≤M2(1+∥g∥0,T,β),
where
(3.9)M2=2M1(1+b1T(1+|x0|)e2M1b1T).In the sequel, we naturally denoteM~1, Δ~1, and M~2 the corresponding quantities that are related to (7).


### 3.2. Proof of Proposition2.3

Proof.
Let0≤s≤t≤T. We write
(3.10)xt-x~t-(xs-x~s)=I1+I2+I3+I4+I5,
where
(3.11)I1=∫st‍[b(xr)-b(x~r)]dr,I2=∫st‍[b(x~r)-b~(x~r)]dr,I3=∫st‍[σ(xr)-σ(x~r)]dgr,I4=∫st‍[σ(x~r)-σ~(x~r)]dgr,I5=∫st‍σ~(x~r)d[gr-g~r].
Since for any s≤r≤t, |xr-x~r|≤∥x-x~∥s,t,β+|xs-x~s|, we may write
(3.12)|I1|≤b1{∥x-x~∥s,t,β(t-s)1+β+|xs-x~s|(t-s)}
and clearly
(3.13)|I2|≤∥b-b~∥∞(t-s).
We use (2.3) to obtain
(3.14)|I4|≤cβ∥g∥0,T,β{∥σ-σ~∥∞(t-s)β+∥σ′-σ′~∥∞∥x~∥s,t,β(t-s)2β},|I5|≤cβ∥g-g~∥0,T,β{∥σ~∥∞(t-s)β+∥σ′~∥∞∥x~∥s,t,β(t-s)2β}.
Then by (3.8) we have for (t-s)≤Δ~1:
(3.15)|I4|≤M3∥g∥0,T,β∥σ-σ~∥C1(t-s)β,|I5|≤M3∥g-g~∥0,T,β∥σ~∥C1(t-s)β,
where M3=cβ(1+b~1T(1+|x~0|)e2M~1b~1T).
The termI3 is a little bit more elaborate. First we have
(3.16)|σ(xr)-σ(x~r)-σ(xr′)+σ(x~r′)|≤∥σ′∥∞|xr-x~r-xr′+x~r′|+∥σ′′∥∞|xr-xr′|(|xr-x~r|+|xr′-x~r′|)≤c1∥x-x~∥s,t,β(t-s)β+2c2∥x∥s,t,β(∥x-x~∥s,t,β(t-s)β+|xs-x~s|)(t-s)β
and thus
(3.17)∥σ(x·)-σ(x~·)∥s,t,β≤c1∥x-x~∥s,t,β+2c2∥x∥s,t,β(∥x-x~∥s,t,β(t-s)β+|xs-x~s|).
Since
(3.18)∥σ(x·)-σ(x~·)∥s,t,∞≤∥σ′∥∞∥x-x~∥s,t,β(t-s)β+∥σ′∥∞|xs-x~s|.
Inequality (2.3) yields
(3.19)|I3|≤cβ∥g∥0,T,β{c1∥x-x~∥s,t,β(t-s)2β+c1|xs-x~s|(t-s)β+[c1∥x-x~∥s,t,β+2c2∥x∥s,t,β(∥x-x~∥s,t,β(t-s)β+|xs-x~s|)](t-s)2β}
and by (3.8) we deduce that for (t-s)≤Δ1:
(3.20)|I3|≤M4∥g∥0,T,β{∥x-x~∥s,t,β(t-s)2β+|xs-x~s|(t-s)β},
with M4=2cβ∥σ∥C1(1+b1T(1+|x0|)e2M1b1T). We use (3.12), (3.13), (3.20), and (3.15), in (3.10) and we may write that for (t-s)≤Δ2:=min(Δ1,Δ~1)(3.21)∥x-x~∥s,t,β≤b1|xs-x~s|(t-s)1-β+M4∥g∥0,T,β|xs-x~s|+∥b-b~∥∞(t-s)1-β+M3∥σ-σ~∥C1∥g∥0,T,β+M3∥σ~∥C1∥g-g~∥0,T,β+∥x-x~∥s,t,β[b1(t-s)+M4∥g∥0,T,β(t-s)β]≤b1|xs-x~s|(t-s)1-β+M4∥g∥0,T,β|xs-x~s|+E+M5∥x-x~∥s,t,β(1+∥g∥0,T,β)(t-s)β,
where
(3.22)E=∥b-b~∥∞(t-s)1-β+M3∥σ-σ~∥C1∥g∥0,T,β+M3∥σ~∥C1∥g-g~∥0,T,β
and M5=max(b1T1-β,M4). With Δ3=(2M5(1+∥g∥0,T,β))-1/β we may write when (t-s)≤Δ:=min(Δ2,Δ3) that
(3.23)∥x-x~∥s,t,β≤2b1|xs-x~s|(t-s)1-β+2M4∥g∥0,T,β|xs-x~s|+2E.
The final arguments are the same as in the proof of (3.1). For t-s≤Δ we have
(3.24)∥x-x~∥s,t,∞≤|xs-x~s|(1+2b1Δ+2M4∥g∥0,T,βΔβ)+2E
that implies by induction if we denote N=T/Δ(3.25)∥x-x~∥0,T,∞≤(|x0-x~0|+2NE)(1+2b1Δ+2M4∥g∥0,T,βΔβ)N≤N(|x0-x~0|+2E)(1+2b1T+2M4T∥g∥0,T,βΔβ-1T/Δ)T/Δ≤(TΔ)(|x0-x~0|+2E)exp{2b1T+2M4T∥g∥0,T,βΔβ-1}.
Now we see that we may find a constant M depending only on T, cβ and all the coefficients of (2.4) (but not on g and g~) such that
(3.26)∥x-x~∥0,T,∞≤C(|x0-x~0|+E)(1+∥g∥0,T,β)1/βexp{C∥g∥0,T,β1/β}.
Substituting the bound (3.26) in (3.23) yields that for (t-s)≤Δ:
(3.27)∥x-x~∥s,t,β≤C(|x0-x~0|+E)(1+∥g∥0,T,β)1+1/βexp{C∥g∥0,T,β1/β}.
Thus by Lemma A.2 from [11] we finally obtain
(3.28)∥x-x~∥0,T,β≤C(|x0-x~0|+E)(1+∥g∥0,T,β)2/βexp{C∥g∥0,T,β1/β}.
Proposition 2.3 is now a consequence of (3.26) and (3.28).


### 3.3. Proof of Theorem1.1

Proof.
The proof is now very simple. We use (2.10) from Proposition 2.3 with g=BH and g~=BH,n. Moreover for any k≥1 we have the following estimate of the moment of the Hölder norm of the fBm (see [16, Lemma 8] for instance)
(3.29)𝔼(∥BH∥0,T,β2k)≤32k(2T)2k(H-β)(2k)!k!.
Then the estimate on the moments of ∥Xn-X∥β are deduced from easy algebra using Lemma 2.2, (3.29), Hölder’s inequality and Young’s inequality. The convergence (1.9) follows from the stability assumptions on the coefficients.


## 3.1. A Preliminary Lemma

We will need the following lemma whose proof is borrowed from [11]. For the sake of completeness and to give some information on the constants that are involved in the statement, we briefly recall the arguments that are used in the proof.Lemma 3.1.
There exists an explicit constantM1 that depends on T, b0, b1, c0, c1, and cβ such that
(3.1)∥x∥0,T,∞≤T(1+|x0|)e2M1b1T(2M1(1+∥g∥0,T,β))1/β.Proof.
Let0≤s≤t≤T. Since |b(xr)|≤|b(xr)-b(xs)|+|b(xs)| and |b(xs)|≤b0+b1|xs|, we have
(3.2)|∫st‍b(xr)dr|≤b0(t-s)+b1|xs|(t-s)+b1∥x∥s,t,β(t-s)1+β.
Clearly ∥σ(x·)∥s,t,β≤c1∥x∥s,t,β, thus Inequality (2.3) yields
(3.3)∥x∥s,t,β≤b0(t-s)1-β+b1|xs|(t-s)1-β+b1∥x∥s,t,β(t-s)+cβ∥g∥0,T,β[c0+c1∥x∥s,t,β(t-s)β]≤M1{1+∥g∥0,T,β+b1|xs|(t-s)1-β+(1+∥g∥0,T,β)∥x∥s,t,β(t-s)β}
with M1=max(1,T1-β∥b∥C1,cβ∥σ∥C1). We denote
(3.4)Δ1={2M1(1+∥g∥0,T,β)}-1/β
and when (t-s)≤Δ1 we may write
(3.5)∥x∥s,t,β≤2M1(1+∥g∥0,T,β+b1|xs|Δ11-β).
Since ∥x∥s,t,∞≤|xs|+∥x∥s,t,β(t-s)β, we obtain for (t-s)≤Δ1 that
(3.6)∥x∥s,t,∞≤|xs|(1+2M1b1Δ1)+1.
By induction with N=T/Δ1 it follows that
(3.7)∥x∥0,T,∞≤|x0|(1+2M1b1Δ1)N+∑k=1N‍(1+2M1b1Δ1)k≤(TΔ1)(1+|x0|)(1+2M1b1Δ1)T/Δ1
and since for any a>0, (1+x/a)a≤ex, we have (1+2M1b1Δ1)T/Δ1≤e2M1b1T and we finally deduce (3.1).
As noticed in [11], if b is the null function, then b1=0 and the last above argument fails. In this case it is impossible to obtain a bound without any exponential of the quantity ∥g∥0,T,β.Remark 3.2.
When(t-s)≤Δ1, we substitute (3.1) into (3.5) and we deduce the estimate
(3.8)∥x∥s,t,β≤M2(1+∥g∥0,T,β),
where
(3.9)M2=2M1(1+b1T(1+|x0|)e2M1b1T).In the sequel, we naturally denoteM~1, Δ~1, and M~2 the corresponding quantities that are related to (7).


## 3.2. Proof of Proposition2.3

Proof.
Let0≤s≤t≤T. We write
(3.10)xt-x~t-(xs-x~s)=I1+I2+I3+I4+I5,
where
(3.11)I1=∫st‍[b(xr)-b(x~r)]dr,I2=∫st‍[b(x~r)-b~(x~r)]dr,I3=∫st‍[σ(xr)-σ(x~r)]dgr,I4=∫st‍[σ(x~r)-σ~(x~r)]dgr,I5=∫st‍σ~(x~r)d[gr-g~r].
Since for any s≤r≤t, |xr-x~r|≤∥x-x~∥s,t,β+|xs-x~s|, we may write
(3.12)|I1|≤b1{∥x-x~∥s,t,β(t-s)1+β+|xs-x~s|(t-s)}
and clearly
(3.13)|I2|≤∥b-b~∥∞(t-s).
We use (2.3) to obtain
(3.14)|I4|≤cβ∥g∥0,T,β{∥σ-σ~∥∞(t-s)β+∥σ′-σ′~∥∞∥x~∥s,t,β(t-s)2β},|I5|≤cβ∥g-g~∥0,T,β{∥σ~∥∞(t-s)β+∥σ′~∥∞∥x~∥s,t,β(t-s)2β}.
Then by (3.8) we have for (t-s)≤Δ~1:
(3.15)|I4|≤M3∥g∥0,T,β∥σ-σ~∥C1(t-s)β,|I5|≤M3∥g-g~∥0,T,β∥σ~∥C1(t-s)β,
where M3=cβ(1+b~1T(1+|x~0|)e2M~1b~1T).
The termI3 is a little bit more elaborate. First we have
(3.16)|σ(xr)-σ(x~r)-σ(xr′)+σ(x~r′)|≤∥σ′∥∞|xr-x~r-xr′+x~r′|+∥σ′′∥∞|xr-xr′|(|xr-x~r|+|xr′-x~r′|)≤c1∥x-x~∥s,t,β(t-s)β+2c2∥x∥s,t,β(∥x-x~∥s,t,β(t-s)β+|xs-x~s|)(t-s)β
and thus
(3.17)∥σ(x·)-σ(x~·)∥s,t,β≤c1∥x-x~∥s,t,β+2c2∥x∥s,t,β(∥x-x~∥s,t,β(t-s)β+|xs-x~s|).
Since
(3.18)∥σ(x·)-σ(x~·)∥s,t,∞≤∥σ′∥∞∥x-x~∥s,t,β(t-s)β+∥σ′∥∞|xs-x~s|.
Inequality (2.3) yields
(3.19)|I3|≤cβ∥g∥0,T,β{c1∥x-x~∥s,t,β(t-s)2β+c1|xs-x~s|(t-s)β+[c1∥x-x~∥s,t,β+2c2∥x∥s,t,β(∥x-x~∥s,t,β(t-s)β+|xs-x~s|)](t-s)2β}
and by (3.8) we deduce that for (t-s)≤Δ1:
(3.20)|I3|≤M4∥g∥0,T,β{∥x-x~∥s,t,β(t-s)2β+|xs-x~s|(t-s)β},
with M4=2cβ∥σ∥C1(1+b1T(1+|x0|)e2M1b1T). We use (3.12), (3.13), (3.20), and (3.15), in (3.10) and we may write that for (t-s)≤Δ2:=min(Δ1,Δ~1)(3.21)∥x-x~∥s,t,β≤b1|xs-x~s|(t-s)1-β+M4∥g∥0,T,β|xs-x~s|+∥b-b~∥∞(t-s)1-β+M3∥σ-σ~∥C1∥g∥0,T,β+M3∥σ~∥C1∥g-g~∥0,T,β+∥x-x~∥s,t,β[b1(t-s)+M4∥g∥0,T,β(t-s)β]≤b1|xs-x~s|(t-s)1-β+M4∥g∥0,T,β|xs-x~s|+E+M5∥x-x~∥s,t,β(1+∥g∥0,T,β)(t-s)β,
where
(3.22)E=∥b-b~∥∞(t-s)1-β+M3∥σ-σ~∥C1∥g∥0,T,β+M3∥σ~∥C1∥g-g~∥0,T,β
and M5=max(b1T1-β,M4). With Δ3=(2M5(1+∥g∥0,T,β))-1/β we may write when (t-s)≤Δ:=min(Δ2,Δ3) that
(3.23)∥x-x~∥s,t,β≤2b1|xs-x~s|(t-s)1-β+2M4∥g∥0,T,β|xs-x~s|+2E.
The final arguments are the same as in the proof of (3.1). For t-s≤Δ we have
(3.24)∥x-x~∥s,t,∞≤|xs-x~s|(1+2b1Δ+2M4∥g∥0,T,βΔβ)+2E
that implies by induction if we denote N=T/Δ(3.25)∥x-x~∥0,T,∞≤(|x0-x~0|+2NE)(1+2b1Δ+2M4∥g∥0,T,βΔβ)N≤N(|x0-x~0|+2E)(1+2b1T+2M4T∥g∥0,T,βΔβ-1T/Δ)T/Δ≤(TΔ)(|x0-x~0|+2E)exp{2b1T+2M4T∥g∥0,T,βΔβ-1}.
Now we see that we may find a constant M depending only on T, cβ and all the coefficients of (2.4) (but not on g and g~) such that
(3.26)∥x-x~∥0,T,∞≤C(|x0-x~0|+E)(1+∥g∥0,T,β)1/βexp{C∥g∥0,T,β1/β}.
Substituting the bound (3.26) in (3.23) yields that for (t-s)≤Δ:
(3.27)∥x-x~∥s,t,β≤C(|x0-x~0|+E)(1+∥g∥0,T,β)1+1/βexp{C∥g∥0,T,β1/β}.
Thus by Lemma A.2 from [11] we finally obtain
(3.28)∥x-x~∥0,T,β≤C(|x0-x~0|+E)(1+∥g∥0,T,β)2/βexp{C∥g∥0,T,β1/β}.
Proposition 2.3 is now a consequence of (3.26) and (3.28).


## 3.3. Proof of Theorem1.1

Proof.
The proof is now very simple. We use (2.10) from Proposition 2.3 with g=BH and g~=BH,n. Moreover for any k≥1 we have the following estimate of the moment of the Hölder norm of the fBm (see [16, Lemma 8] for instance)
(3.29)𝔼(∥BH∥0,T,β2k)≤32k(2T)2k(H-β)(2k)!k!.
Then the estimate on the moments of ∥Xn-X∥β are deduced from easy algebra using Lemma 2.2, (3.29), Hölder’s inequality and Young’s inequality. The convergence (1.9) follows from the stability assumptions on the coefficients.



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*Source: 281474-2012-12-05.xml*