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

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Abstract

Given a bounded domain ΩRd and two integro-differential operators L1, L2 of the form Lju(x)=p.v.Ω(u(x)u(y))kj(x,y,xy)dy we study the fully nonlinear Bellman equation(0.1)maxj=1,2{Lju(x)+aj(x)u(x)fj(x)}=0in Ω, with Dirichlet boundary conditions. Here, aj,fj:ΩR are non-negative functions. We prove the existence of a non-negative function u:ΩR which satisfies (0.1) almost everywhere. The main difficulty arises through the nonlocality of Lj and the absence of regularity near the boundary.

MSC

35J60
47G20
60J75
93E20

Keywords

Bellman equation
Fully nonlinear equation
Integro-differential operator
Markov jump process
Stochastic control

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Partially supported by DFG (German Science Foundation) through SFB 611.