# The Mean Field Games

• Alain Bensoussan
• Jens Frehse
• Phillip Yam
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

## Abstract

Let us set
$$\displaystyle{ a(x) = \frac{1} {2}\sigma {(x)\sigma }^{{\ast}}(x), }$$
(3.1)
and introduce the second-order differential operator
$$\displaystyle{ A\varphi (x) = -\text{tr }a(x){D}^{2}\varphi (x). }$$
(3.2)
We define the dual operator
$$\displaystyle{ {A}^{{\ast}}\varphi (x) = -\sum _{ k,l=1}^{n} \frac{{\partial }^{2}} {\partial _{x_{k}}\partial _{x_{l}}}(a_{kl}(x)\varphi (x)). }$$
(3.3)
Since m(t) is the probability distribution of $$\hat{x}(t)$$, it has a density with respect to the Lebesgue measure denoted by m(x, t), which is the solution of the Fokker–Planck equation
$$\displaystyle\begin{array}{rcl} \frac{\partial m} {\partial t} + {A}^{{\ast}}m + \text{div }(g(x,m,\hat{v}(x))m)& =& 0, \\ m(x,0)& =& m_{0}(x).{}\end{array}$$
(3.4)
We next want the feedback $$\hat{v}(x)$$ to solve a standard control problem, in which m appears as a parameter. We can thus readily associate an HJB equation with this problem, parametrized by m.

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© Alain Bensoussan, Jens Frehse, Phillip Yam 2013

## Authors and Affiliations

• Alain Bensoussan
• 1
• 2
• Jens Frehse
• 3
• Phillip Yam
• 4
1. 1.Naveen Jindal School of ManagementUniversity of Texas at DallasRichardsonUSA
2. 2.Department of Systems Engineering and Engineering ManagementCity University of Hong KongKowloonHong Kong SAR
3. 3.Institut für Angewandte MathematikUniversitat BonnBonnGermany
4. 4.Department of StatisticsThe Chinese University of Hong KongShatinHong Kong SAR