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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 \(1/2 \leq \theta \leq 1\), and they are exponentially mean-square stable if the stepsize \(\Delta t \in (0, \Delta t_0)\) when \(0 \leq \theta < 1\). Finally, some numerical experiments are given to illustrate the theoretical results.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)

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