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MiS Preprint

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

Helmut Abels and Moritz Kassmann


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.

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

Related publications

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