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://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/790.html
KW - Poisson jump, Lipschitz condition, semi-implicit Euler method, exponential stability, convergence.
AB - <p style="text-align: justify;">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 $p$th moment for some $p &gt; 2$ and give the convergence result.</p>