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

In this paper, we study the mean square stability of the solution and its stochastic theta scheme for the following stochastic differential equations driven by fractional Brownian motion with Hurst parameter H∈(12,1): dX(t)=f(t,X(t))dt+g(t,X(t))dBH(t). Firstly, we consider the special case when f(t,X)=−λκtκ−1X and g(t,X)=μX. Secondly, the stability of the solution and its stochastic theta scheme for nonlinear equations is studied. Due to presence of long memory, even the problem of stability in the mean square sense of the solution has not been well studied, let alone the stability of numerical schemes. A complete new set of techniques to deal with this difficulty are developed. Numerical examples are carried out to illustrate our theoretical results.