Numerical approximation of SDEs driven by fractional Brownian motion for all $H\in(0,1)$ using WIS integration
<|reference_start|>Numerical approximation of SDEs driven by fractional Brownian motion for all $H\in(0,1)$ 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 $H\in(0,1)$. 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.<|reference_end|>
@article{erdogan2024numerical,
    title={Numerical approximation of SDEs driven by fractional Brownian motion for
  all $H\in(0,1)$ using WIS integration},
    author={Utku Erdogan and Gabriel J. Lord and Roy B. Schieven},
    journal={arXiv preprint arXiv:2404.07013},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.07013},
    primaryClass={math.NA cs.NA}
}