Brownian Particles Controlled by Their Occupation Measure
Copyright policy )Brownian Particles Controlled by Their Occupation Measure
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.
Hamilton-Jacobi-Bellman equations, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Brownian particles, Hamilton-Jacobi equations in optimal control and differential games, Probability (math.PR), calculus on the space of measures, HJB equations, Stochastic optimal control, Calculus on the space of measures, Linear-quadratic optimal control problems, FOS: Mathematics, Optimal stochastic control, occupation measure, Boué-Dupuis formula, 93E20, 49J55, 60G57, 49L12, stochastic optimal control, Mathematics - Probability, Occupation measure, Random measures
Hamilton-Jacobi-Bellman equations, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Brownian particles, Hamilton-Jacobi equations in optimal control and differential games, Probability (math.PR), calculus on the space of measures, HJB equations, Stochastic optimal control, Calculus on the space of measures, Linear-quadratic optimal control problems, FOS: Mathematics, Optimal stochastic control, occupation measure, Boué-Dupuis formula, 93E20, 49J55, 60G57, 49L12, stochastic optimal control, Mathematics - Probability, Occupation measure, Random measures
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