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Stability analysis between the hybrid stochastic delay differential equations with jumps and the Euler-Maruyama method
Guangjie Li,Qigui Yang
Keywords:Mean square stability, Stochastic delay differential equations, Euler-Maruyama method, Stability equivalence, Markovian switching, Jumps.
Abstract:
      The aim of this paper is to concern with the mean square exponential stability equivalence between the hybrid stochastic delay differential equations with jumps and the Euler-Maruyama method (EM-method). Precisely, under the global Lipschitz condition, it is shown that a stochastic delay differential equation with Markovian switching and jumps (SDDEwMJ) is mean square exponentially stable if and only if for some sufficiently small step size, its EM-method is mean square exponentially stable. Based on such a result, the mean square exponential stability of a SDDEwMJ can be investigated by the careful numerical simulations in practice without resorting to Lyapunov functions. Moreover, a numerical example is provided to confirm the obtained results.