Solutions of martingale problems for Lévy-type operators with discontinuous coefficients and related SDEs

Peter Imkeller and Niklas Willrich

Stochastic Processes and their Applications, 2016, vol. 126, issue 3, 703-734

Abstract: We show the existence of Lévy-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 null set on which the singularities of the limit operator are located.

Keywords: Lévy process; Stable process; Stable-like process; Lévy-type process; Discontinuous Lévy characteristics; Non-local operator; Markov process; Feller process; Symbol; Martingale problem; Weak solution; Stochastic differential equation; Skorokhod space; Pseudo-differential operator (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:126:y:2016:i:3:p:703-734

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DOI: 10.1016/j.spa.2015.09.017

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