Mean-square Stability and Convergence of Compensated Split-Step $theta$-method for Nonlinear Jump Diffusion Systems

Document Type : Original Article

Authors

1 Department of applied mathematics Ferdowsi university of Mashhad Mashhad, Iran

2 Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

3 Department of Statistics, Ferdowsi University of Mashhad, Mashhad,

10.22054/jmmf.2020.54500.1011

Abstract

In this paper, we analyze the strong convergence and stability of the Compensated Splite-step $theta$ (CSS$theta$) and Forward-Backward Euler-Maruyama (FBEM) methods for Numerical solutions of Stochastic Differential Equations with jumps (SDEwJs),
where ‎$sqrt{2}-1leqthetaleq 1‎$. The drift term $f$ has a one-sided Lipschitz condition, the diffusion term $g$ and jump term $h$ satisfy global Lipschitz condition.
Furthermore, we discuss about the stability of SDEwJs with constant coefficients and present new useful relations between their coefficients. Finally we examine the correctness and efficiency of theorems with some examples.
In this paper, we analyze the strong convergence and stability of the Compensated Splite-step $theta$ (CSS$theta$) and Forward-Backward Euler-Maruyama (FBEM) methods for Numerical solutions of Stochastic Differential Equations with jumps (SDEwJs),
where ‎$sqrt{2}-1leqthetaleq 1‎$. The drift term $f$ has a one-sided Lipschitz condition, the diffusion term $g$ and jump term $h$ satisfy global Lipschitz condition.
Furthermore, we discuss about the stability of SDEwJs with constant coefficients and present new useful relations between their coefficients. Finally we examine the correctness and efficiency of theorems with some examples.

Keywords