Pathwise stochastic control and a class of stochastic partial differential equations

TL;DR

This paper investigates a pathwise stochastic optimal control problem and establishes the uniqueness of the solution to the associated Hamilton-Jacobi-Bellman SPDE, also exploring a stochastic pathwise Noether theorem.

Contribution

It introduces a novel approach to solving pathwise stochastic control problems and proves the uniqueness of solutions to the related SPDE in the viscosity sense.

Findings

Value process is the unique viscosity solution of the SPDE.

Established a stochastic pathwise Noether theorem.

Provided insights into non-adapted Hamilton-Jacobi-Bellman equations.

Abstract

We consider a pathwise stochastic optimal control problem and study the associated (not necessarily adapted) Hamilton-Jacobi-Bellman stochastic partial differential equation. We show that the value process is the unique solution of this equation, in the viscosity sense. Finally, we discuss a version of some corresponding stochastic pathwise Noether theorem.