Second Order Hamilton--Jacobi Equations in Hilbert Spaces and Stochastic Boundary Control

Abstract

The paper is concerned with fully nonlinear second order Hamilton--Jacobi--Bellman--Isaacs equations of elliptic type in separable Hilbert spaces which have unbounded first and second order terms. The viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved. A stochastic optimal control problem driven by a parabolic stochastic PDE with control of Dirichlet type on the boundary is considered. It is proved that the value function of this problem is the unique viscosity solution of the associated Hamilton--Jacobi--Bellman equation.