An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control (Q879373)

The authors prove and discuss the existence of a nonnegative solutions \(u: \Omega \to \mathbb{R}_{+}\) to the fully nonlinear Bellman equation \[ \max_{j=1,2}\left\{L^{j} u(x)+a^{j}(x)u(x)-f^{j}(x)\right\}=0 \] in \(\Omega\) with Dirichlet boundary conditions where \(\Omega \subset\mathbb{R}^{d}\) is a bounded domian, \(a^{j}, f^{j}: \Omega \to \mathbb{R}\) are nonnegative functions, and \[ L^{j}u(x) = p.v.\int_{\Omega}(u(x)-u(y))k^{j}(x,y,x-y)\,dy \] are nonlocal integro-differential operators of order \(\alpha \in (1,2).\)