Abstract
Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod's selection theorem.
Highlights
Consider a fractional Brownian motion, a self-similar Gaussian process with stationary increments. It was introduced by Kolmogorov [5] and studied by Mandelbrot and Van Ness [6]
We give some properties of an fractional Brownian motion (fBm), definitions, and some tools used in the proofs
We introduce the linear operator KH∗ from ζ to L2([0, T ]) defined by
Summary
Consider a fractional Brownian motion (fBm), a self-similar Gaussian process with stationary increments. It was introduced by Kolmogorov [5] and studied by Mandelbrot and Van Ness [6]. The fBm with Hurst parameter H ∈ (0, 1) is a centered Gaussian process with covariance function. Has a unique strong solution, which will be assumed throughout this paper. Notice that if the drift coefficient is Lipschitz continuous, Eq (1) has a unique strong solution, which is continuous with respect to the initial condition. Our purpose in this paper is to establish some stability results under the pathwise uniqueness of solutions and under weak regularity conditions on the drift coefficient b.
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