| Citation: |
Guangjie Li, Qigui Yang. STABILITY ANALYSIS BETWEEN THE HYBRID STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH JUMPS AND THE EULER-MARUYAMA METHOD[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1259-1272. doi: 10.11948/20200127
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STABILITY ANALYSIS BETWEEN THE HYBRID STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH JUMPS AND THE EULER-MARUYAMA METHOD
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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.
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References
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Guangjie Li, Qigui Yang. STABILITY ANALYSIS BETWEEN THE HYBRID STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH JUMPS AND THE EULER-MARUYAMA METHOD[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1259-1272. doi: 10.11948/20200127
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Figure 1. The numerical simulation of the
$ r(t) $ (left) and the$ N(t) $ (right). - Figure 2. The mean square curves of numerical solution (left) and the Lyapunov exponent (right) of Eq. (4.1).