Asymptotic preserving schemes for SDEs driven by fractional Brownian motion in the averaging regime

Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1 2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that the...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The c...

Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion

In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H . We consider two cases. If H > 1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in...

A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one

In this paper, we will focus in dimension one on the SDEs of the type dXt = σ(Xt)dB H t + b(Xt)dt where B H is a fractional Brownian motion. Our principal motivation is to describe one of the simplest theory from our point of view allowing to study this SDE, and this for any H ∈ (0, 1). We will consider several definitions of solution and we will study, for each one of them, in which condition ...