Necessary and sufficient conditions for optimal control of semilinear stochastic partial differential equations

Using a recently introduced representation of the second order adjoint state as the solution of a function-valued backward stochastic partial differential equation (SPDE), we calculate the viscosity super- and subdifferential of the value function evaluated along an optimal trajectory for controlled semilinear SPDEs. This establishes the well-known connection between Pontryagin's maximum principle and dynamic programming within the framework of viscosity solutions. As a corollary, we derive that the correction term in the stochastic Hamiltonian arising in nonsmooth stochastic control problems is nonpositive. These results directly lead us to a stochastic verification theorem for fully nonlinear Hamilton–Jacobi–Bellman equations in the framework of viscosity solutions.