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

We study the following stochastic differential delay equations driven by Poisson random jump measuredX(t)=f(X(t),X(t-τ(t)))dt+g(X(t),X(t-τ(t)))dW(t)+∫Rnh(X(t),X(t-τ(t)),u)N∼(dt,du),0⩽t⩽T,where time delay τ(t)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 12∧γ in the mean-square sense.