Existence of (Markovian) solutions to martingale problems associated with L\'evy-type operators

TL;DR

This paper establishes conditions for the existence of solutions to martingale problems associated with Lévy-type operators, including those with discontinuous coefficients, and demonstrates their applications to stochastic differential equations and non-local operators.

Contribution

It provides new existence results for martingale problems with discontinuous coefficients and introduces a Markovian selection theorem ensuring strong Markov processes.

Findings

Existence of solutions for martingale problems with discontinuous symbols.

Application to Lévy-driven SDEs with measurable coefficients.

Establishment of a Harnack inequality for non-local operators of variable order.

Abstract

Let $A$ be a pseudo-differential operator with symbol $q(x,\xi)$. In this paper we derive sufficient conditions which ensure the existence of a solution to the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem. If the symbol $q$ depends continuously on the space variable $x$, then the existence of solutions is well understood, and therefore the focus lies on martingale problems for pseudo-differential operators with discontinuous coefficients. We prove an existence result which allows us, in particular, to obtain new insights on the existence of weak solutions to a class of L\'evy-driven SDEs with Borel measurable coefficients and on the the existence of stable-like processes with discontinuous coefficients. Moreover, we establish a Markovian selection theorem which shows that - under mild assumptions - the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem gives rise to a strong Markov process. The result applies, in particular, to L\'evy-driven SDEs. We illustrate the Markovian selection theorem with applications in the theory of non-local operators and equations; in particular, we establish under weak regularity assumptions a Harnack inequality for non-local operators of variable order.