Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

Abstract

This paper deals with the almost sure exponential stability of the Eulertype methods for nonlinear stochastic delay differential equations with jumps by using the discrete semimartingale convergence theorem. It is shown that the explicit Euler method reproduces the almost sure exponential stability under an additional linear growth condition. By replacing the linear growth condition with the one-sided Lipschitz condition, the backward Euler method is able to reproduce the stability property.

Numerical methods for a class of jump–diffusion systems with random magnitudes

“…Thus there will be small intervals of time where the discretization approach acquires the wrong jump magnitude. The result obtained improves and generalizes some results in [6,[8][9][10][11]14]. The paper is organized as follows.…”

“…Recently, there is extensive literature on the numerical simulation of SDDEs without jumps, and efforts are now being made to bring jump SDDEs up a similar level. In [6][7][8][9][10] strong convergence and mean-square stability properties were analyzed in the case of Poisson-driven jumps of deterministic magnitude. However, it is more common and significant in the financial models for the case where jump magnitudes are random [11][12][13].…”

Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

“…Actually, we only use condition (6) on f. But in the following theorem, which ensures the exponential mean-square stability of the θ -method, we cannot relax the linear growth condition on f, the reason being that the θ -method (when θ = 0) may fail to preserve the stability of the trivial solution of Equation (1) (see Higham et al, 2007;Li & Gan, 2011;Wu et al, 2010). Now, we show that the θ -method can preserve the exponential mean-square stability of the trivial solution of Equation (1) under more strengthened conditions.…”

“…Stochastic θ -method includes the commonly used EM method and backward Euler-Maruyama (BEM) method by choosing θ = 0 and θ = 1, and it is more general than these two methods (Cao, Liu, & Fan, 2004;Higham & Kloeden, 2005;Higham, Mao, & Yuan, 2007;Kloeden & Platen, 1992;Li & Cao, 2015;Li & Gan, 2011;Tan & Wang, 2011;Wang, Mei, & Xue, 2007;Wu, Mao, & Szpruch, 2010). In recent decades, stochastic θ -method has been increasingly used to cope with NSDDEs and stochastic delay differential equations (SDDEs) with jumps.…”

Mean‐square stability of the backward Euler–Maruyama method for neutral stochastic delay differential equations with jumps

“…Meanwhile, in order to build more realistic models to describe the phenomenon of discontinuous random pulse excitation coming up from the underlying systems, jumps as key features have been incorporated into NSDDEs. The models with jumps have become more popular in finance and several areas of science and engineering . In this paper, we will consider a more general model which contains both neutral term and Poisson jumps, named NSDDEs with jumps.…”