Causal Hamilton-Jacobi-Bellman Equations for Anticipative Stochastic Optimal Control
Optimization and Control
2025-10-07 v2
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.
Cite
@article{arxiv.2507.08657,
title = {Causal Hamilton-Jacobi-Bellman Equations for Anticipative Stochastic Optimal Control},
author = {Peter Bank and Franziska Bielert},
journal= {arXiv preprint arXiv:2507.08657},
year = {2025}
}