Convergence of numerical solutions to stochastic differential delay equations with Poisson jump and Markovian switching

In general stochastic delay different equations with Poisson jump and Markovian switching do not have explicit solutions. Appropriate numerical approximations, such as the Euler scheme, are therefore a vital tool in exploring their properties. Unfortunately, the numerical methods for stochastic delay differential equations with Poisson jump and Markovian switching (SDDEwPJMSs), have never been studied. In this paper we proved that the Euler approximate solutions will converge to the exact solutions for SDDEwPJMSs under local Lipschitz condition. This result is more general than what they deal with the Markovian switching term or the jump term.