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,Λ). The process (X,Λ), called a regime-switching jump diffusion with Lévy type jumps, is further shown to possess Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.