Volume 6, Issue 4
Numerical Solutions of Stochastic Differential Delay Equations with Jumps
DOI:

Int. J. Numer. Anal. Mod., 6 (2009), pp. 659-679

Published online: 2009-06

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• Abstract

In this paper, the semi-implicit Euler (SIE) method for the stochastic differential delay equations with Poisson jump and Markov switching (SDDEwPJMSs) is developed. We show that under global Lipschitz assumptions the numerical method is convergent and SDDEwPJMSs is exponentially stable in mean-square if and only if for some sufficiently small step-size $Delta$ the SIE method is exponentially stable in mean-square. We then replace the global Lipschitz conditions with local Lipschitz conditions and the assumptions that the exact and numerical solution have a bounded pth moment for some p > 2 and give the convergence result.

• Keywords

Poisson jump Lipschitz condition semi-implicit Euler method exponential stability convergence

65C30 65L20 60H10

@Article{IJNAM-6-659, author = {G. Zhao, M. Song and M. Liu}, title = {Numerical Solutions of Stochastic Differential Delay Equations with Jumps}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2009}, volume = {6}, number = {4}, pages = {659--679}, abstract = {In this paper, the semi-implicit Euler (SIE) method for the stochastic differential delay equations with Poisson jump and Markov switching (SDDEwPJMSs) is developed. We show that under global Lipschitz assumptions the numerical method is convergent and SDDEwPJMSs is exponentially stable in mean-square if and only if for some sufficiently small step-size $Delta$ the SIE method is exponentially stable in mean-square. We then replace the global Lipschitz conditions with local Lipschitz conditions and the assumptions that the exact and numerical solution have a bounded pth moment for some p > 2 and give the convergence result. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/790.html} }
TY - JOUR T1 - Numerical Solutions of Stochastic Differential Delay Equations with Jumps AU - G. Zhao, M. Song & M. Liu JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 659 EP - 679 PY - 2009 DA - 2009/06 SN - 6 DO - http://dor.org/ UR - https://global-sci.org/intro/ijnam/790.html KW - Poisson jump KW - Lipschitz condition KW - semi-implicit Euler method KW - exponential stability KW - convergence AB - In this paper, the semi-implicit Euler (SIE) method for the stochastic differential delay equations with Poisson jump and Markov switching (SDDEwPJMSs) is developed. We show that under global Lipschitz assumptions the numerical method is convergent and SDDEwPJMSs is exponentially stable in mean-square if and only if for some sufficiently small step-size $Delta$ the SIE method is exponentially stable in mean-square. We then replace the global Lipschitz conditions with local Lipschitz conditions and the assumptions that the exact and numerical solution have a bounded pth moment for some p > 2 and give the convergence result.