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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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
Keywords
- Stochastic differential delay equation
- Polynomial growth condition
- Strong convergence
- Backward Euler–Maruyama method
- Almost surely exponential stability
- Discrete semi-martingale convergence theorem