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

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• 作者：Peter Imkeller^a ; Niklas Willrich^b ; ^{willrich@wias-berlin.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60J25 ; 60H10 ; 60J75 ; 60G46 ; 60G52 ; 47G30
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：126
• 期：3
• 页码：703-734
• 全文大小：339 K

文摘

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.