Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus

TL;DR

This paper develops a discrete Wick calculus-based approximation method for solutions of linear SDEs driven by fractional Brownian motion with Hurst parameter H in (1/2,1), including geometric cases, using Donsker-type approximations and Wick difference equations.

Contribution

It introduces a novel approximation scheme for fractional SDEs using discrete Wick calculus and binary random walks, extending previous methods to geometric fractional Brownian motion.

Findings

Successfully approximates solutions of fractional SDEs in Wick sense

Provides a discrete scheme based on binary random walks and Wick difference equations

Establishes convergence via Hermite recursion formula approximation

Abstract

We approximate the solution of some linear systems of SDEs driven by a fractional Brownian motion $B^H$ with Hurst parameter $H\in(\frac{1}{2},1)$ in the Wick--It\^{o} sense, including a geometric fractional Brownian motion. To this end, we apply a Donsker-type approximation of the fractional Brownian motion by disturbed binary random walks due to Sottinen. Moreover, we replace the rather complicated Wick products by their discrete counterpart, acting on the binary variables, in the corresponding systems of Wick difference equations. As the solutions of the SDEs admit series representations in terms of Wick powers, a key to the proof of our Euler scheme is an approximation of the Hermite recursion formula for the Wick powers of $B^H$.