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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 θ methods inherit the stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize Δt=τ/m when 1/2θ1, and they are exponentially mean-square stable if the stepsize Δt(0,Δt 0 ) when 0θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.
60H10Stochastic ordinary differential equations
65C30Stochastic differential and integral equations
60H35Computational methods for stochastic equations