Causal Hamilton-Jacobi-Bellman Equations for Anticipative Stochastic Optimal Control

TL;DR

This paper develops a novel Hamilton-Jacobi-Bellman framework for stochastic optimal control problems where the controller anticipates future noise, using rough differential equations and functional derivatives to handle the anticipative nature.

Contribution

It introduces a new HJB equation incorporating anticipativity via Dupire's functional derivatives and combines martingale optimality with Itô's formula for such problems.

Findings

Derivation of a new HJB equation for anticipative control.

Integration of rough differential equations into stochastic control.

Application of functional derivatives in the HJB framework.

Abstract

We consider a stochastic optimal control problem where the controller can anticipate the evolution of the driving noise over some dynamically changing time window. The controlled state dynamics are understood as a rough differential equation. We combine the martingale optimality principle with a functional form of It\^o's formula to derive a Hamilton-Jacobi-Bellman (HJB) equation for this problem. This HJB equation is formulated in terms of Dupire's functional derivatives and involves a transport equation arising from the anticipativity of the problem.