The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels

revised version: October 2007
Helmut Abels and Moritz Kassmann
(Please use for correspondence this email).

Submission date: 05. Oct. 2006
Pages: 30
published in: Osaka journal of mathematics, 46 (2009) 3, p. 661-683 
MSC-Numbers: 47G20, 47G30, 60J75, 60J35, 60G07, 35K99, 35B65, 47A60
Keywords and phrases: martingale problem, cauchy problem, pseudodifferential operator, Levy-type process, jump process
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Abstract:
We consider the linear integro-differential operator L defined by

Here the kernel k(x,y) behaves like , , for small y and is Hölder-continuous in the first variable, precise definitions are given below. The aim of this work is twofold. On one hand, we study the unique solvability of the Cauchy problem corresponding to L. On the other hand, we study the martingale problem for L. The analytic results obtained for the deterministic parabolic equation guarantee that the martingale problem is well-posed. Our strategy follows the classical path of Stroock-Varadhan. The assumptions allow for cases that have not been dealt with so far.