Stochastic Control and Differential Games with Path-Dependent Influence of Controls on Dynamics and Running Cost

TL;DR

This paper develops a path-dependent Hamilton-Jacobi-Bellman framework for stochastic control and differential games where controls influence dynamics and costs based on their entire past trajectory, extending classical methods.

Contribution

It introduces a path-dependent HJB equation and proves a Dynamic Programming Principle for control problems with path-dependent controls and costs.

Findings

Derived a path-dependent HJB equation using functional Itô calculus.

Proved a Dynamic Programming Principle for path-dependent control problems.

Applied the framework to delay-type path dependence and differential games.

Abstract

In this paper, we consider the functional It\^o calculus framework to find a path-dependent version of the Hamilton-Jacobi-Bellman equation for stochastic control problems that feature dynamics and running cost that depend on the path of the control. We also prove a Dynamic Programming Principle for such problems. We apply our results to path-dependence of the delay type. We further study Stochastic Differential Games in this context.