On the Martingale Problem and Feller and Strong Feller Properties for Weakly Coupled L\'evy Type Operators

TL;DR

This paper establishes the well-posedness and Feller properties of a regime-switching jump diffusion process driven by Le9vy type operators, using coupling methods under mild and non-Lipschitz conditions.

Contribution

It proves the well-posedness and Feller properties of a class of weakly coupled Le9vy type operators and their associated Markov processes.

Findings

Martingale problem is well-posed and uniquely determines the process.

The process has Feller and strong Feller properties under non-Lipschitz conditions.

Coupling method is effective for proving regularity properties.

Abstract

This paper considers the martingale problem for a class of weakly coupled L\'{e}vy 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 L\'evy type jumps, is further shown to posses Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.