Convergence of jump-diffusion non-linear differential equation with phase semi-Markovian switching

Semi-Markov process (SMP) is a generalization of Markov process, which can overcome the restriction of the negative exponential distribution of the sojourn time at a state. In this paper, our focus is on the class of jump-diffusion stochastic delay differential equation with phase semi-Markovian switching. By employing the theta method, we prove that, for p ⩾ 2, the pth-moment of discrete and continuous approximation solutions are stable, and the pth-moment error of the continuous approximation solution is convergent under some weak conditions. The techniques of proof are quite general and hence have the potential to be applied to other numerical models.