Approximations of Fractional Stochastic Differential Equations by Means of Transport Processes

Details Approximations of Fractional Stochastic Differential Equations by Means of Transport Processes Approximations of Fractional Stochastic Differential Equations by Means of Transport Processes

2011-02-24

arxiv.org/abs/1102.5067v2

Mathematics

Probability

Scientific paper

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.

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