db(t),\\
r(T)=-\bar{Q}_{T}S_{T}y(T);
\end{array}
\right.$$ and $$u(x,t)=U(x,m(t),t)=\frac{1}{2}x^{*}P(t)x+x^{*}r(t)+s(t).$$

We have $$s(t)=\frac{1}{2}y^{*}\Gamma(t)y+\mu(t).$$

Since $\Gamma(t),y(t),\mu(t)$ are solutions of equations identical to the Mean Field type control, we again obtain $$\begin{aligned}
s(t) & =\frac{1}{2}y(T)^{*}S_{T}^{*}\bar{Q}_{T}S_{T}y(T)+\int_{t}^{T}(\frac{1}{2}\text{tr}\, aP(s)+\frac{\beta^{2}}{2}\text{tr}P(s)+\beta^{2}\text{tr }\Sigma(s))ds\nonumber \\
&\qquad+ \int_{t}^{T}(\frac{1}{2}y(s)^{*}S^{*}QSy(s)-\frac{1}{2}r(s)^{*}BR^{-1}B^{*}r(s)+r(s)^{*}\bar{A}y(s))ds-\beta\int_{t}^{T}y(s)^{*}\Gamma(s)db(s).\nonumber \end{aligned}$$

The result coincides with that in Bensoussan [@BSYY] when $\beta=0$.

NASH EQUILIBRIUM FOR A FINITE NUMBER OF PLAYERS 
================================================

THE PROBLEM FOR A FINITE NUMBER OF PLAYERS
------------------------------------------

We consider here $N$ players. Each of them has a state $x^{i}(t)\in \mathbb{R}^{n}$. We define the vector of states $x(t)\in \mathbb{R}^{nN}$ as $$x(t)=(x^{1}(t),\cdots,x^{N}(t)).$$

We denote $x_{l}^{i}(t),l=1,\cdots,n,$ the components of the state $x^{i}(t).$ The states evolve according to the following dynamics $$\left\{
\begin{array}{l}
dx^{i}(s) =g(x,v^{i}(x))ds+\sigma(x^{i})dw^{i}(s)+\beta db(s),\; s>t,\label{eq:6.1}\\
x^{i}(t) =x^{i};
\end{array}
\right.$$ in which $v^{i}(x)\in \mathbb{R}^{d}$ is the control of player $i$. The processes $w^{i}(s)$ and $b(s)$ are independent standard Wiener processes in $\mathbb{R}^{n}$. We set, as usual, $a(x^{i})=\sigma(x^{i})\sigma(x^{i})^{*}$. It is convenient to denote $v(x)=(v^{1}(x),\cdots,v^{N}(x)).$ We introduce the cost functional of each player $$J^{i}(x,t;v(\cdot))= \mathbb{E} [\int_{t}^{T}f(x(s),v^{i}(x(s))ds+h(x(T))]\label{eq:6.2}.$$

We notice that the trajectories and cost functionals are linked only through the states and not through the controls. Consider the Hamiltonian $$H(x,q)=\inf_{v}[f(x,v)+q.g(x,v)].\label{eq:6.3}$$

Denote $\hat{v}(x,q)$ the minimizer in the Hamiltonian, then we set $G(x,q)=g(x,\hat{v}(x,q))$. We next consider the system of PDEs, $i=1, \cdots, 