Mean square stability of stochastic theta method for stochastic differential equations driven by fractional Brownian motion

TL;DR

This paper investigates the mean-square stability of solutions and stochastic theta numerical schemes for stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 0.5, revealing conditions under which stability is preserved.

Contribution

It provides new theoretical insights and techniques for analyzing the stability of stochastic theta schemes for fractional Brownian motion driven equations, including nonlinear cases.

Findings

Stability depends on the parameter  and the Hurst parameter H.

The stochastic theta method reproduces stability under certain parameter conditions.

Numerical examples confirm the theoretical stability results.

Abstract

In this paper, we study the mean-square stability of the solution and its stochastic theta scheme for the following stochastic differential equations drive by fractional Brownian motion with Hurst parameter $H\in (\frac 12,1)$: $$ dX(t)=f(t,X(t))dt+g(t,X(t))dB^{H}(t). $$ Firstly, we consider the special case when $f(t,X)=-\lambda\kappa t^{\kappa-1}X$ and $g(t,X)=\mu X$. The solution is explicit and is mean-square stable when $\kappa\geq 2H$. It is proved that if the parameter $2H\leq\kappa\le 3/2$ and $\frac{\sqrt{3/2}\cdot e}{\sqrt{3/2}\cdot e+1} (\approx 0.77)\leq \theta\leq 1$ or $\kappa>3/2$ and $1/2<\theta\le 1$, the stochastic theta method reproduces the mean-square stability; and that if $0<\theta<\frac 12$, the numerical method does not preserve this stability unconditionally. Secondly, we study the stability of the solution and its stochastic theta scheme for nonlinear equations. Due to the presence of long memory, even the problem of stability in the mean square sense of the solution has not been well studied since the conventional techniques powerful for stochastic differential equations driven by Brownian motion are no longer applicable. Let alone the stability of numerical schemes. We need to develop a completely new set of techniques to deal with this difficulty. Numerical examples are carried out to illustrate our theoretical results.