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 H？lder 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 -H？lder continuous paths for all (see [1] 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 H？lder continuous of order strictly larger than 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 -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 Z？hle [5]). Using this formula Nualart and R？？canu 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 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 H？lder 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