# On Martingale Problems and Feller Processes

## TL;DR

This paper links the well-posedness of certain martingale problems to the existence of Feller processes, providing new results for Lévy-driven SDEs and stable-like processes with unbounded coefficients.

## Contribution

It establishes a sufficient condition connecting martingale problem well-posedness to Feller processes and applies this to new existence and uniqueness results for complex stochastic models.

## Key findings

- Martingale problem well-posedness implies Feller process existence.
- New existence and uniqueness results for Lévy-driven SDEs.
- Results applicable to stable-like processes with unbounded coefficients.

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