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Brownian particles controlled by their occupation measure

Loïc Béthencourt, Rémi Catellier, Etienne Tanré

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 Itô 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 Boué-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.

Brownian particles controlled by their occupation measure

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 Itô 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 Boué-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.
Paper Structure (11 sections, 12 theorems, 126 equations)

This paper contains 11 sections, 12 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Calculus on the space of measures
  3. Basics on the space of finite measures
  4. Chain-rule formula for the flow of finite measures
  5. Stochastic calculus and occupation measures
  6. A general Itô formula
  7. Related formulas for diffusion processes
  8. The control problem
  9. The Boué-Dupuis formula.
  10. The HJB equation.
  11. Application: controlling the volume of Brownian particles

Key Result

Proposition 3

Let $u\in \mathrm{C}^1(\mathscr{M}_c^d)$, then for any $(\mu, x)\in \mathscr{M}_c^d\times \mathbb{R}^d$, it holds that As a consequence, if $u\in \mathrm{C}^1(\mathscr{M}_c^d)$, then its linear derivative is uniquely defined.

Theorems & Definitions (35)

  • Definition 1
  • proof
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Remark 5
  • Proposition 6
  • proof
  • ...and 25 more