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

TL;DR

This paper tackles numerical approximation of quasilinear SDEs driven by fractional Brownian motion for all Hurst parameters $H\in(0,1)$ using the Wick-It\^o-Skorohod (WIS) integral. It introduces the GBMEM integrating-factor method, builds on the Wiener-It\^o chaos/WIS framework, and proves a strong convergence rate of $(\mathbb{E}[|X(t_n)-X_n|^2])^{1/2} \le C_H \Delta t^{\min(H,\zeta)}$ for the general non-autonomous setting, with numerical evidence suggesting higher rates in autonomous cases (near $\Delta t^{H}$ or $\Delta t^{H+1/2}$). The work combines rigorous WIS-based analysis (including Gjessing's Lemma and translation operators) with practical algorithms for simulating SDEs across all $H$, and demonstrates that accurate, unbiased simulations are feasible even for small $H$. The results have potential impact on applications requiring robust SDE simulations with fractional noise, spanning finance, physics, and engineering, by providing a concrete, implementable scheme that respects the WIS interpretation of the integral. Open questions include improving the theoretical convergence rate to match numerically observed values and extending the approach to multi-dimensional or nonlinear diffusion terms.

Abstract

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 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 $O(Δt^H)$ and in the non-autonomous case $O(Δt^{\min(H,ζ)})$, where $ζ$ is a 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.

Figures (6)

• Figure 1: Sample paths of GBMEM for different values of $H$ using same random numbers for \ref{['eq: quasi-linear SDE']} with $\beta=1$, $a(x)=\frac{4x}{1+x^2}$, $x_0=1$, $\Delta t=0.001$ and (a) $\alpha=0$ (MishuraEM) (b) $\alpha=1$ (GBMEM).
• Figure 2: Sample paths of the four methods with $\alpha=1$, $\beta=1$, $a(x)=\frac{4x}{1+x^2}$, $x_0=1$, $\Delta t=0.001$ using same random numbers with (a) $H=0.25$, (b) $H=0.75$.
• Figure 3: The estimated RMSE of the four methods for various values of $\Delta t$ for the quasi-linear SDE with parameters in \ref{['eq: num ex']}, $\alpha=1$ and $T=1$, including a linear fit for the GBMEM method. We have (a) $H=0.25$, linear fit slope $0.771$ (b) $H=0.75$, linear fit slope $1.009$.
• Figure 4: Estimated rates of convergence for $\beta=1$, $a(x)=\frac{4x}{1+x^2}$, $x_0=1$ and $T=1$. The total Monte Carlo sample size is $500$, divided in batches of $50$ to obtain the given error bars. In (a) $\alpha=0$ (MishuraEM) and in (b) $\alpha=1$ (GBMEM).
• Figure 5: (a) The estimated rates of convergence for $\alpha=1$, $\beta=2$, $a(x)=\frac{4x}{1+x^2}$, $x_0=1$ and $T=1$ (GBMEM). (b) The estimated rates of convergence for $\alpha=-1$, $\beta=0.5$, $a(x)=\cos(x)$, $x_0=10$ and $T=1$ (GBMEM). In (a) and (b) the total Monte Carlo sample size is $500$, divided in batches of $50$ to obtain the given error bars.

Theorems & Definitions (32)

• Theorem 2.1: Wiener--Itô chaos expansion
• Proposition 2.2
• proof
• Remark 2.3
• Proposition 2.4
• Corollary 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.8
• Theorem 2.9: Wick chain rule