Convergence of numerical solutions for variable delay differential equations driven by Poisson random jump measure

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.