Numerical approximation of SDEs driven by fractional Brownian motion for all H∈(0,1) using WIS integration

We examine the numerical approximation of a quasilinear stochastic differential equation (SDE) with multiplicative fractional Brownian motion. The stochastic integral is interpreted in the Wick-It\^o-Skorohod (WIS) sense that is well defined and centered for all H∈(0,1). We give an introduction to the theory of WIS integration before we examine existence and uniqueness of a solution to the SDE. We then introduce our numerical method which is based on the theoretical results in {Mishura2008article, Mishura2008} for H≥ 1/2. We construct explicitly a translation operator required for the practical implementation of the method and are not aware of any other implementation of a numerical method for the WIS SDE. We then prove a strong convergence result that gives, in the non-autonomous case, an error of O(Δ tH) and in the non-autonomous case O(Δ tmin(H,ζ)), where ζ is a H\"older continuity parameter. We present some numerical experiments and conjecture that the theoretical results may not be optimal since we observe numerically a rate of min(H+1/2,1) in the autonomous case. This work opens up the possibility to efficiently simulate SDEs for all H values, including small values of H when the stochastic integral is interpreted in the WIS sense.