Approximation of stochastic differential equations driven by subfractional Brownian motion at discrete time observation

Abstract

In this paper, we consider discrete time approximations for stochastic differential equations with the form: $$X_t=X_0+{\int }_0^{t}f(X_s)d{h}_s+{\int }_0^{t}g(X_s)d{Y}_s^H,\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t>0,$$ where $h:{\mathbb{R}}^{+}\to \mathbb{R}$ is a continuous function with locally bounded variation, $f,g:\mathbb{R}\to \mathbb{R}$ are measurable functions, and the integral with respect to $Y_t^H={\int }_0^t{\sigma }_{s}d{S}_s^H$ is the pathwise Riemann-Stieltjes integral, S^H is a subfractional Brownian motion with $H\in (\frac{1}{2},1),$ σ is a deterministic (possibly discontinuous) function.

Keywords:

• Subfractional Brownian motion
• Gaussian processes
• discrete time approximation

2000 MSC:

• 60F17
• 60G15
• 60G18
• 60G22

Acknowledgments

The authors would like to thank anonymous referees and editor whose remarks and suggestions greatly improved the presentation of our paper.

Additional information

Funding

Guangjun Shen is supported by the National Natural Science Foundation of China (Grant No. 12071003), the Distinguished Young Scholars Foundation of Anhui Province (Grant No. 1608085J06). Jun Wang is supported by the National Natural Science Foundation of China (Grant No.11971432).