Abstract
We study the following stochastic differential delay equations driven by Poisson random jump measure dX ( t ) = f ( X ( t ) , X ( t - τ ( t ) ) ) dt + g ( X ( t ) , X ( t - τ ( t ) ) ) dW ( t ) + ∫ R n h ( X ( t ) , X ( t - τ ( t ) ) , u ) N ∼ ( dt , du ) , 0 ⩽ t ⩽ T , where time delay τ ( t ) is a variant and N ∼ ( dt , du ) is a compensated Poisson random measure. In this paper, the semi-implicit Euler approximate solutions are established and we show the convergence of numerical approximate solutions to the true solutions; Further we prove that the semi-implicit Euler method is convergent with order 1 2 ∧ γ in the mean-square sense.
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