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Strong convergence and stability of backward Euler–Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation

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Abstract

In the paper, our main aim is to investigate the strong convergence and almost surely exponential stability of an implicit numerical approximation under one-sided Lipschitz condition and polynomial growth condition on the drift coefficient, and polynomial growth condition on the diffusion coefficient. After providing almost surely exponential stability and moment boundedness for the exact solution, we show that an appropriate implicit numerical method preserves boundedness of moments, and the numerical approximation converges strongly to the true solution. Moreover, we prove that the backward Euler–Maruyama approximation can share almost surely exponential stability of the exact solution for sufficiently small step size.

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Acknowledgments

The author expresses her sincere gratitude to three anonymous referees for their detailed comments and helpful suggestions. The financial support from the National Natural Science Foundation of China (Grant No. 11301198) and Fundamental Research Funds for the Central Universities (2011QN167).

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  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Shaobo Zhou

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  1. Shaobo Zhou
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Zhou, S. Strong convergence and stability of backward Euler–Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation. Calcolo 52, 445–473 (2015). https://doi.org/10.1007/s10092-014-0124-x

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  • DOI: https://doi.org/10.1007/s10092-014-0124-x

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