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On martingale problems and Feller processes. (English) Zbl 1390.60278
Summary: 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.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 60G44 Martingales with continuous parameter 60J75 Jump processes (MSC2010) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G51 Processes with independent increments; Lévy processes
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