Research Article | Open Access
A Stability Result for Stochastic Differential Equations Driven by Fractional Brownian Motions
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.
1. Introduction and Main Result
Suppose that is an -dimensional fractional Brownian motion (fBm in short) with Hurst parameter defined on a complete filtered probability space . We mean that the components , are independent centered Gaussian processes with the covariance function If , then is clearly a Brownian motion. Since for any , the processes have -Hölder continuous paths for all (see  for further information about fBm).
In this paper we fix and we consider the solution of the following stochastic differential equation (abbreviated by SDE from now on) on , is the initial value of the process .
Under suitable assumptions on , the processes and have trajectories which are Hölder continuous of order strictly larger than so we can use the integral introduced by Young in . 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  using the notion of -variation. The theory of rough paths introduced by Lyons in  was used by Coutin and Qian in order to prove an existence and uniqueness result for (1.2) (see ). The Riemann-Stieltjes integral appearing in (1.2) can be expressed as a Lebesgue integral using a fractional integration by parts formula (see Zähle ). Using this formula Nualart and Răşcanu have established in  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 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  (see also ), 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 function from to , we denote (if the following quantities do exist) and The space is the space of continuously differentiable functions such that . The space is defined in a similar way. For and , we denote by the space of -Hölder continuous functions , equipped with the norm where We simply write when .
The main result of this work is the following theorem.
Theorem 1.1. Let be a sequence of fractional Brownian motions defined on . Let , , and be some sequences in , , and . One considers the sequence of processes such that for any , is the unique solution of If there exists(i), and such that (ii)a fractional Brownian motion defined on such that for and one has then where is the solution of (1.2) with the coefficients , , and and driven by the fBm .
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, Theorem 1.1 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 two deterministic rough functions and in and two deterministic differential equations where all the coefficients are smooth. Then we will prove that there exists a constant such that Since our upper bound is explicit, this estimate can be viewed as a refinement of Theorems 11.3 and 11.6 of . 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 . The stability with respect to all the coefficients in (1.2) is new to our knowledge.
The paper is organized as follows. In Section 2 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 .
Suppose that and with . From , the Riemann-Stieltjes integral exists. In , the author provides an explicit expression for the integral in terms of fractional derivatives. In order to give some precisions, we consider being such that . Supposing that the following limit exists and is finite, we denote . Then the Riemann-Stieltjes integral can be expressed as where We refer to  for further details on fractional operators. The following useful lemma is now classical. Its proof is postponed in the appendix.
Lemma 2.1. Let and in with and , then there exists a universal constant such that
Set and let . We will work with the following deterministic differential equations on : for and .
We introduce the following assumptions on the coefficients of the above equations. For a function from to , denotes the matrix of first order derivatives and denotes its Hessian.(H1) There exists some positive constants , , , such that , , , and . (H2) There exists some positive constants , , , , , and such that , , , , , and . It is proved in [6, Theorem 5.1] that if , each of the above equations has a unique -Hölder continuous solution. The estimates on the solution obtained in  were improved in . Let us recall the following observations concerning Theorem 3 in Hu and Nualart  (see also  for similar comments). It has been proved in  that if and is twice continuously differentiable with bounded second order derivatives, then there exists a constant that depends on , , and such that Replacing and by the trajectories of the fractional Brownian motions and , we obtain estimation in the supremum norm for the difference of the processes and satisfying 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. Let , . Then for any
One refers to  for a proof of this lemma. Lemma 2.2 implies the following integrability property for (or ): for any and , Inequality (2.8) together with (2.5) will unfortunately be useless since the quantities and appear in a multiplicative way in the exponent in the right hand side of (2.5). Indeed, Young’s inequality yields and if one imposes then necessarily and the finiteness of the expression such as 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 when and are the solutions of (2.4).
Proposition 2.3. Let be fixed and let and be Hölder continuous of order . Under (H1) and (H2), there exists a constant that depends only on , , , , , , , , , and such that
It is worth to notice that a careful reading of the proof shows that depends continuously on its parameters. This is important when we apply this proposition to stability properties of stochastic differential equations.
The subject of this section is the proof of Proposition 2.3 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 for simplicity. Thus for a function , we denote its derivative and its second order derivative. Moreover designates the constant in (2.3).
3.1. A Preliminary Lemma
We will need the following lemma whose proof is borrowed from . 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 constant that depends on , , , , , and such that
Proof. Let . Since and , we have
Clearly , thus Inequality (2.3) yields
with . We denote
and when we may write
Since , we obtain for that
By induction with it follows that
and since for any , , we have and we finally deduce (3.1).
As noticed in , if is the null function, then and the last above argument fails. In this case it is impossible to obtain a bound without any exponential of the quantity .
In the sequel, we naturally denote , , and the corresponding quantities that are related to (7).
3.2. Proof of Proposition 2.3
Proof. Let . We write
Since for any , , we may write
We use (2.3) to obtain
Then by (3.8) we have for :
The term is a little bit more elaborate. First we have and thus Since Inequality (2.3) yields and by (3.8) we deduce that for : with . We use (3.12), (3.13), (3.20), and (3.15), in (3.10) and we may write that for where and . With we may write when that The final arguments are the same as in the proof of (3.1). For we have that implies by induction if we denote Now we see that we may find a constant depending only on , and all the coefficients of (2.4) (but not on and ) such that Substituting the bound (3.26) in (3.23) yields that for : Thus by Lemma from  we finally obtain Proposition 2.3 is now a consequence of (3.26) and (3.28).
3.3. Proof of Theorem 1.1
Proof. The proof is now very simple. We use (2.10) from Proposition 2.3 with and . Moreover for any we have the following estimate of the moment of the Hölder norm of the fBm (see [16, Lemma 8] for instance) Then the estimate on the moments of 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.
Proof of Lemma 2.1
Proof. With , we use (2.1) and we obtain for all and all : We have It follows that We use the change of variables and we recall that the Beta function is defined by . Then we get It is proved in  that in fact where is a universal constant.
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