Approximations of Fractional Stochastic Differential Equations by Means of Transport Processes

Abstract

We present strong approximations with rate of convergence for the solution of a stochastic differential equation of the form $$ dX_t=b(X_t)dt+σ(X_t)dB^H_t, $$ where $b\in C^1_b$, $σ\in C^2_b$, $B^H$ 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 Garzón et al (2009). If $σ$ is bounded away from 0, an approximation is obtained by a general Lipschitz dependence result of Römisch and Wakolbinger (1985). Without that assumption on $σ$, 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.