Optimal Control of Unbounded Stochastic Evolution Systems in Hilbert Spaces

TL;DR

This paper develops a new theory of viscosity solutions for second-order Hamilton-Jacobi-Bellman equations in Hilbert spaces, addressing unbounded stochastic systems without requiring B-continuity of coefficients.

Contribution

Introduces a novel notion of viscosity solution for HJB equations in Hilbert spaces that does not rely on B-continuity, extending the existing theory.

Findings

Proves the value functional is the unique continuous viscosity solution.

Shows the new viscosity solution coincides with classical solutions.

Removes the B-continuity assumption on coefficients in the theory.

Abstract

Optimal control and the associated second-order Hamilton-Jacobi-Bellman (HJB) equation are studied for unbounded stochastic evolution systems in Hilbert spaces. A new notion of viscosity solution, featured by absence of B-continuity, is introduced for the second-order HJB equation in the sense of Crandall and Lions, and is shown to coincide with the classical solutions and to satisfy a stability property. The value functional is proved to be the unique continuous viscosity solution to the second-order HJB equation, with the coefficients being not necessarily B-continuous. Our result provides a new theory of viscosity solutions to the HJB equation for optimal control of stochastic evolutionary equations-driven by a linear unbounded operator-in a Hilbert space, and removes the B-continuity assumption on the coefficients which is used in the existing literature.