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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.

Suppose that

In this paper we fix

Under suitable assumptions on

In order to obtain moment bounds on the solution of (

Now we present the kind of results we are interested in and so we need further notations. For a differentiable function

The main result of this work is the following theorem.

Let

a fractional Brownian motion

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

The paper is organized as follows. In Section

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 (

Suppose that

Let

Set

We introduce the following assumptions on the coefficients of the above equations. For a function

There exists some positive constants

There exists some positive constants

Let

One refers to [

Therefore the next result is a strengthening of Theorem 3.3 in [

Let

It is worth to notice that a careful reading of the proof shows that

The subject of this section is the proof of Proposition

We will need the following lemma whose proof is borrowed from [

There exists an explicit constant

Let

As noticed in [

When

In the sequel, we naturally denote

Let

The term

The proof is now very simple. We use (

With