On The Interpretation Of The Master Equation

TL;DR

This paper explores the interpretation and various formulations of the Master Equation, focusing on its origins, different approaches, and the recent proposal to work within the Hilbert space of square integrable random variables.

Contribution

It provides a comprehensive discussion of the Master Equation's interpretations, origins, and the recent Hilbert space approach introduced by P.L. Lions.

Findings

Analyzes multiple perspectives on the Master Equation.

Discusses the origin and modeling aspects of the equation.

Highlights the recent Hilbert space formulation by Lions.

Abstract

Since its introduction by P.L. Lions in his lectures and seminars at the College de France, see [9], and also the very helpful notes of Cardialaguet [4] on Lions' lectures, the Master Equation has attracted a lot of interest, and various points of view have been expressed, see for example Carmona-Delarue [5], Bensoussan-Frehse-Yam [2], Buckdahn-Li-Peng-Rainer [3]. There are several ways to introduce this type of equation; and in those mentioned works, they involve an argument which is a probability measure, while P.L. Lions has recently proposed the idea of working with the Hilbert space of square integrable random variables. Hence writing the equation is an issue; while another issue is its origin. In this article, we discuss all these various aspects, and our modeling argument relies heavily on a seminar at College de France delivered by P.L. Lions on November 14, 2014.