Stochastic control with rough paths

TL;DR

This paper develops a framework for controlled rough differential equations, establishing a Hamilton-Jacobi-Bellman equation and Pontryagin maximum principle, extending duality theory to continuous time via rough paths.

Contribution

It introduces a novel formulation of controlled rough differential equations, generalizing Rogers' duality formula from discrete to continuous time using rough path theory.

Findings

Value function satisfies a HJB type equation

Established a Pontryagin maximum principle for rough differential equations

Linked to classical duality results in stochastic control

Abstract

We study a class of controlled rough differential equations. It is shown that the value function satisfies a HJB type equation; we also establish a form of the Pontryagin maximum principle. Deterministic problems of this type arise in the duality theory for controlled diffusion processes and typically involve anticipating stochastic analysis. We propose a formulation based on rough paths and then obtain a generalization of Roger's duality formula [L. C. G. Rogers, 2007] from discrete to continuous time. We also make the link to old work of [Davis--Burstein, 1987].