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

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  • Imkeller, Peter
  • Willrich, Niklas

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.

Suggested Citation

  • Imkeller, Peter & Willrich, Niklas, 2016. "Solutions of martingale problems for Lévy-type operators with discontinuous coefficients and related SDEs," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 703-734.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:3:p:703-734
    DOI: 10.1016/j.spa.2015.09.017
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    References listed on IDEAS

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    1. Bass, Richard F. & Tang, Huili, 2009. "The martingale problem for a class of stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1144-1167, April.
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