Approximations of Fractional Stochastic Differential Equations by Means of Transport Processes

TL;DR

This paper develops strong approximation methods with convergence rates for solutions of fractional stochastic differential equations using transport processes and Euler schemes, extending existing techniques to cases where the diffusion coefficient may approach zero.

Contribution

It introduces new approximation techniques for fractional SDEs, especially handling cases with non-constant diffusion coefficients near zero, based on transport processes and Euler schemes.

Findings

Established strong convergence rates for approximations of fractional SDEs.

Extended approximation methods to cases where the diffusion coefficient is not bounded away from zero.

Provided detailed proofs and conditions for the convergence of the proposed schemes.

Abstract

We present strong approximations with rate of convergence for the solution of a stochastic differential equation of the form $$ dX_t=b(X_t)dt+\sigma(X_t)dB^H_t, $$ where $b\in C^1_b$, $\sigma \in C^2_b$, $B^H$ is fractional Brownian motion with Hurst index $H$, and we assume existence of a unique solution with Doss-Sussmann representation. The results are based on a strong approximation of $B^H$ by means of transport processes of Garz\'on et al (2009). If $\sigma$ is bounded away from 0, an approximation is obtained by a general Lipschitz dependence result of R\"omisch and Wakolbinger (1985). Without that assumption on $\sigma$, that method does not work, and we proceed by means of Euler schemes on the Doss-Sussmann representation to obtain another approximation, whose proof is the bulk of the paper.