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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
15/2006

An Analytic Approach to Purely Nonlocal Bellman Equations Arising in Models of Stochastic Control

Helmut Abels and Moritz Kassmann

Abstract

Given a bounded domain $\Omega \subset \mathbb{R}^d$ and two integro-differential operators $L^1, L^2$ of the form $L^j u(x) = p.v. \int_{\Omega} (u(x)-u(y))k^j(x,y,x-y)dy$ we study the fully nonlinear Bellman equation \begin{align} \max\limits_{j=1,2} \big\{ L^j u(x) + a^j(x)u(x) - f^j(x) \big\} &= 0 \quad \text{ in }\ \Omega \ (0.1) \end{align} with Dirichlet boundary conditions. Here, $a^j, f^j: \Omega \to \mathbb{R}$ are nonnegative functions. We prove the existence of a nonnegative function $u:\Omega \to \mathbb{R}$ which satisfies $(0.1)$ almost everywhere. The main difficulty arises through the nonlocality of $L^j$ and the absence of regularity near the boundary.

Received:
Feb 10, 2006
Published:
Feb 10, 2006
MSC Codes:
35J60, 47G20, 60J75, 93E20
Keywords:
Bellman equation, fully nonlinear equation, ntegro-differential operator, Markov jump process, stochastic control

Related publications

inJournal
2007 Repository Open Access
Helmut Abels and Moritz Kassmann

An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control

In: Journal of differential equations, 236 (2007) 1, pp. 29-56