Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations

Abstract

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 non-smooth stochastic control problems is non-positive. These results directly lead us to a stochastic verification theorem for fully nonlinear Hamilton--Jacobi--Bellman equations in the framework of viscosity solutions.

To appear at Ann. Appl. Probab

References in corpus (2)

• Peng's Maximum Principle for Stochastic Partial Differential Equations
• Stochastic Verification Theorem for Infinite Dimensional Stochastic Control Systems

Cited by in corpus (1)

• Stochastic optimal control in Hilbert spaces: $C^{1,1}$ regularity of the value function and optimal synthesis via viscosity solutions