Search

MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint
109/2006

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

Helmut Abels and Moritz Kassmann

Abstract

We consider the linear integro-differential operator $L$ defined by \[ Lu(x) =\int_{\mathbb{R}^n} \left(u(x+y)-u(x)-\mathbb{1}_{[1,2]}(\alpha)\mathbb{1}_{\{|y|\leq 2\}}(y)y\cdot \nabla u(x) \right)k(x,y) d y \,. \] Here the kernel $k(x,y)$ behaves like $|y|^{-d-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is H\"older-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.

Received:
Oct 5, 2006
Published:
Oct 5, 2006
MSC Codes:
47G20, 47G30, 60J75, 60J35, 60G07, 35K99, 35B65, 47A60
Keywords:
martingale problem, cauchy problem, pseudodifferential operator, Levy-type process, jump process

Related publications

inJournal
2009 Journal Open Access
Helmut Abels and Moritz Kassmann

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

In: Osaka journal of mathematics, 46 (2009) 3, pp. 661-683