relatif à ce cas, et obtenons à nouveau l’équation maitresse.
address:
- |
    International Center for Decision and Risk Analysis\
    Jindal School of Management, University of Texas at Dallas,\
    College of Science and Engineering\
    Systems Engineering and Engineering Management\
    City University Hong Kong
- 'Institute for Applied Mathematics, University of Bonn'
- 'Department of Statistics, The Chinese University of Hong Kong'
author:
- Alain Bensoussan
- Jens Frehse
- Sheung Chi Phillip Yam
title: THE MASTER EQUATION IN MEAN FIELD THEORY
---

Master Equation,Mean Field Type Control Problems,Mean Field Games,Stochastic Maximum Principle,Stochastic HJB Equations,Linear Quadratic Problems

35B37 ,49N70 ,60H15 ,91A06

INTRODUCTION
============

Since we do not intend to give complete proofs, we proceed formally, by assuming relevant smoothness structure whenever it eases the argument. We consider functions $f(x,m,v),\; g(x,m,v),\; h(x,m)$ and $\sigma(x)$ where $x\in \mathbb{R}^{n};$ $m$ is a probability measure on $\mathbb{R}^{n}$, but we shall retain ourselves mostly in the regular case, in which $m$ represents the probability density, assumed to be in $L^{2}(\mathbb{R}^{n})$ and $v$ is a control in $\mathbb{R}^{d}$. The functions $f$ and $h$ are scalar, $g$ is a vector in $\mathbb{R}^{n}$ and $\sigma(x)$ is a $n\times n$ matrix. All these functions are differentiable in all arguments. In 