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

The Bellman equation arises in stochastic control theory when one is looking for the best possible choice of a control function which minimises a cost functional associated to a stochastic process. On the analytic side this is described by a fully non-linear elliptic equation. When the control of Wiener processes is considered, usual second order elliptic differential operators occur in the equation. But considering Markov jump processes non-local integro-differential operators enter into the equation. These operators are similar to fractional powers of the Laplacian and obey a maximum principle, which allows to prove existence of positive solutions of the corresponding Bellman equation.