  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 L\'evy-driven stochastic differential
equations and stable-like processes with unbounded coefficients.
