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 $H\in(0,1)$. 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 \cite{Mishura2008article, Mishura2008} for $H\geq \frac{1}{2}$. 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 $O(\Delta t^H)$ and in the non-autonomous case $O(\Delta t^{\min(H,\zeta)})$, where $\zeta$ 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 $\min(H+\frac{1}{2},1)$ 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.

翻译：本文研究带有乘性分数布朗运动的拟线性随机微分方程（SDE）的数值逼近问题。采用Wick-Itô-Skorohod（WIS）意义上的随机积分，该积分对于所有$H\in(0,1)$均具有良定义性与中心化性质。在探讨该随机微分方程解的存在唯一性之前，我们首先介绍WIS积分理论。随后基于文献\cite{Mishura2008article, Mishura2008}中针对$H\geq \frac{1}{2}$情形建立的理论成果，提出一种数值方法。我们显式构造了方法实际实现所需的平移算子——据我们所知，目前尚无其他针对WIS型随机微分方程的数值方法实现。进而证明强收敛性结果：非自治情形下误差阶为$O(\Delta t^H)$，自治情形下误差阶为$O(\Delta t^{\min(H,\zeta)})$，其中$\zeta$为Hölder连续性参数。通过数值实验发现，在自治情形下数值观察到的收敛率为$\min(H+\frac{1}{2},1)$，据此推测理论结果可能并非最优。本研究为所有$H$值（包括采用WIS积分解释随机积分时的极小$H$值）情形下随机微分方程的高效模拟开辟了新途径。