On the approximations of solutions to neutral SDEs with Markovian switching and jumps under non-Lipschitz conditions

In this paper, we investigate the existence and uniqueness of solutions to neutral stochastic differential equations with Markovian switching and jumps (NSDEwMSJs) under non-Lipschitz conditions. On the other hand, we present the Euler approximate solutions for NSDEwMSJs and show that the convergence of the Euler approximate solutions to the true solutions by applying Ito^ formula, Bihari's lemma and Burkholder-Davis-Gundy's lemma. Some examples are provided to illustrate the main results.