Solutions of martingale problems for L\'evy-type operators and stochastic differential equations driven by L\'evy processes with discontinuous coefficients
Probability
2012-08-09 v1
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.
Keywords
Cite
@article{arxiv.1208.1665,
title = {Solutions of martingale problems for L\'evy-type operators and stochastic differential equations driven by L\'evy processes with discontinuous coefficients},
author = {Peter Imkeller and Niklas Willrich},
journal= {arXiv preprint arXiv:1208.1665},
year = {2012}
}