Control Problem on Space of Random Variables and Master Equation

TL;DR

This paper explores a control problem in the space of random variables, linking the Hamilton-Jacobi-Bellman equation to the Master equation in mean field theory, and compares different mathematical frameworks for this relationship.

Contribution

It introduces a control problem in the space of random variables and compares Hilbert space and Wasserstein metric approaches to the Master equation.

Findings

Establishes the connection between the HJB equation and the Master equation.

Provides extensions to the Wasserstein gradient approach.

Clarifies the relationship between different mathematical frameworks.

Abstract

We study in this paper a control problem in a space of random variables. We show that its Hamilton Jacobi Bellman equation is related to the Master equation in Mean field theory. P.L. Lions in [14,15] introduced the Hilbert space of square integrable random variables as a natural space for writing the Master equation which appears in the mean field theory. W. Gangbo and A. \'Swi\k{e}ch [10] considered this type of equation in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient. We compare the two approaches and provide some extension of the results of Gangbo and \'Swi\k{e}ch.