Brownian particles controlled by their occupation measure
Probability
2025-04-21 v2
Abstract
In this article, we study a finite horizon linear-quadratic stochastic control problem for Brownian particles, where the cost functions depend on the state and the occupation measure of the particles. To address this problem, we develop an It\^o formula for the flow of occupation measure, which enables us to derive the associated Hamilton-Jacobi-Bellman equation. Then, thanks to a Feynman-Kac formula and the Bou\'e-Dupuis formula, we construct an optimal strategy and an optimal trajectory. Finally, we illustrate our result when the cost-function is the volume of the sausage associated to the particles.
Cite
@article{arxiv.2404.06960,
title = {Brownian particles controlled by their occupation measure},
author = {Loïc Béthencourt and Rémi Catellier and Etienne Tanré},
journal= {arXiv preprint arXiv:2404.06960},
year = {2025}
}