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arXiv · 2404.06960

Brownian particles controlled by their occupation measure

Loïc Béthencourt·Rémi Catellier·Etienne Tanré
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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.

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Loïc Béthencourt, Rémi Catellier, Etienne Tanré. 2024-04-10. Brownian particles controlled by their occupation measure. https://doi.org/10.1137/24m1656220

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