Solutions of martingale problems for L\'evy-type operators and stochastic differential equations driven by L\'evy processes with discontinuous coefficients

TL;DR

This paper establishes the existence of one-dimensional Le9vy-type stochastic processes with discontinuous or stable-like coefficients by formulating and solving associated martingale problems with non-local operators.

Contribution

It introduces a novel approach to construct Le9vy-type processes with discontinuous coefficients via approximation of non-local operators, extending the theory to more irregular cases.

Findings

Existence of Le9vy-type processes with discontinuous coefficients.

Construction via approximation of non-local operators.

Processes converge uniformly outside null sets with singularities.

Abstract

We show the existence of L\'evy-type stochastic processes in one space dimension with characteristic triplets that are either discontinuous at thresholds, or are stable-like with stability index functions for which the closures of the discontinuity sets are countable. For this purpose, we formulate the problem in terms of a Skorokhod-space martingale problem associated with non-local operators with discontinuous coefficients. These operators are approximated along a sequence of smooth non-local operators giving rise to Feller processes with uniformly controlled symbols. They converge uniformly outside of increasingly smaller neighborhoods of a Lebesgue nullset on which the singularities of the limit operator are located.