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Brownian particles controlled by their occupation measure

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arxiv 2404.06960 v2 pith:IDLQND2D submitted 2024-04-10 math.PR

Loïc Béthencourt , Rémi Catellier , Etienne Tanré This is my paper · ORCID
classification math.PR
keywords particlesformulameasureoccupationassociatedbrownianoptimalproblem
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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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  1. Controlled Occupied Processes and Viscosity Solutions

    math.OC 2024-11 conditional novelty 7.0 of 10

    A comparison principle is proved for occupied PDEs, the dynamic programming equations of control problems whose state records the occupation measure of a diffusion, giving uniqueness of the viscosity solution.

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