Numerical approximation of SDEs driven by fractional Brownian motion for all $H\in(0,1)$ using WIS integration
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2024
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| author | Erdogan, Utku Lord, Gabriel J. Schieven, Roy B. |
| author_facet | Erdogan, Utku Lord, Gabriel J. Schieven, Roy B. |
| contents | 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-Itô-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 previous theoretical results 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 autonomous case, an error of $O(Δt^H)$ and in the non-autonomous case $O(Δt^{\min(H,ζ)})$, where $ζ$ is a time-Hölder 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical approximation of SDEs driven by fractional Brownian motion for all $H\in(0,1)$ using WIS integration Erdogan, Utku Lord, Gabriel J. Schieven, Roy B. Numerical Analysis 65C30, 60H35, 60G22 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-Itô-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 previous theoretical results 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 autonomous case, an error of $O(Δt^H)$ and in the non-autonomous case $O(Δt^{\min(H,ζ)})$, where $ζ$ is a time-Hölder 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. |
| title | Numerical approximation of SDEs driven by fractional Brownian motion for all $H\in(0,1)$ using WIS integration |
| topic | Numerical Analysis 65C30, 60H35, 60G22 |
| url | https://arxiv.org/abs/2404.07013 |