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Stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps. (English) Zbl 1236.60055
Summary: This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps. A sufficient condition for the mean-square exponential stability of the exact solution is derived. Then, the mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic $\theta$ methods inherit the stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize ${\Delta }t=\tau /m$ when $1/2\le \theta \le 1$, and they are exponentially mean-square stable if the stepsize ${\Delta }t\in \left(0,{\Delta }{t}_{0}\right)$ when $0\le \theta <1$. Finally, some numerical experiments are given to illustrate the theoretical results.
MSC:
 60H10 Stochastic ordinary differential equations 65C30 Stochastic differential and integral equations 60H35 Computational methods for stochastic equations