Optimal control of infinite-dimensional differential systems with randomness and path-dependence and stochastic path-dependent Hamilton-Jacobi equations

TL;DR

This paper develops a stochastic optimal control framework for infinite-dimensional systems with path-dependence and randomness, characterizing the value function via a novel stochastic path-dependent Hamilton-Jacobi equation and establishing its uniqueness as a viscosity solution.

Contribution

It introduces a new stochastic path-dependent Hamilton-Jacobi equation for infinite-dimensional systems and proves the uniqueness of the viscosity solution as the value function.

Findings

Value function is a random field on the path space.

The value function uniquely solves the stochastic path-dependent Hamilton-Jacobi equation.

Extension of deterministic path-dependent control to stochastic infinite-dimensional systems.

Abstract

This paper is devoted to the stochastic optimal control problem of infinite-dimensional differential systems allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases studied by Bayraktar and Keller [J. Funct. Anal. 275 (2018), 2096--2161], the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.