Convergence analysis of semi-implicit Euler methods for solving stochastic equations with variable delays and random Jump magnitudes

In this paper, we study the following stochastic equations with variable delays and random jump magnitudes: {dX(t)=f(X(t),X(t−τ(t)))dt+g(X(t),X(t−τ(t)))dW(t)+h(X(t),X(t−τ(t)),γN(t)+1)dN(t),0≤t≤T,X(t)=ψ(t),−r≤t≤0. We establish the semi-implicit Euler approximate solutions for the above systems and prove that the approximate solutions converge to the analytical solutions in the mean-square sense as well as in the probability sense. Some known results are generalized and improved.