On the Martingale Problem and Feller and Strong Feller Properties for Weakly Coupled Lévy Type Operators

Abstract

This paper considers the martingale problem for a class of weakly coupled Lévy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process $(X,\Lambda )$. The process $(X,\Lambda )$, called a regime-switching jump diffusion with Lévy type jumps, is further shown to posses Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.