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

Given a bounded domain and two integro-differential operators L1, L2 of the form we study the fully nonlinear Bellman equation with Dirichlet boundary conditions. Here, are non-negative functions. We prove the existence of a non-negative function which satisfies (0.1) almost everywhere. The main difficulty arises through the nonlocality of Lj and the absence of regularity near the boundary.