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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.
