On martingale problems and Feller processes

Abstract

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty }(\mathbb{R} ^d))$-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for Lévy-driven stochastic differential equations and stable-like processes with unbounded coefficients.