&\qquad +\eta^{*}\Gamma(t,m_{1})\xi+x^{*}\frac{\partial^{2}}{\partial m_{1}^{2}}\Sigma(t,m_{1})y+\frac{1}{2}y^{*}\frac{\partial^{2}}{\partial m_{1}^{2}}\Gamma(t,m_{1})y+\frac{\partial^{2}}{\partial m_{1}^{2}}\mu(t,m_{1}),\\
    \sum_{i=1}^{n}\frac{\partial}{\partial\xi_{i}}\frac{\partial}{\partial\eta_{i}}(\frac{\partial^{2}U(x,m,t)}{\partial m^{2}}(\xi,\eta))&=\text{tr}\Gamma(t,m_{1}),\\
    \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}\frac{\partial}{\partial\xi_{i}}\frac{\partial U(x,m,t)}{\partial m}(\xi)&=\text{tr}\Sigma(t,m_{1}).
\end{split}$$

We apply these formulas in the Master equation (\[eq:5.31\]) to obtain $$\begin{array}{l}
    -(\displaystyle\frac{1}{2}x^{*}\frac{d}{dt}P(t)x+x^{*}\frac{d}{dt}\Sigma(t,m_{1})y+\frac{1}{2}y^{*}\frac{d}{dt}\Gamma(t,m_{1})y+\frac{d}{dt}\mu(t,m_{1}))-(\frac{1}{2}\text{tr} aP(t)+\frac{1}{2}\beta^{2}\text{tr} P(t))\\
    \qquad-(\Sigma^{*}(t,m_{1})x+\Gamma(t,m_{1})y)^{*}(Ay+\bar{A}ym_{1}-BR^{-1}B^{*}(Py+\Sigma ym_{1}))\\
    \qquad-\displaystyle\frac{\beta^{2}}{2}\text{tr}\Gamma(t,m_{1})m_{1}^{2}-\beta^{2}\text{tr}\Sigma(t,m_{1})m_{1}\\
    =\displaystyle\frac{1}{2}x^{*}(Q+\bar{Q})x-x^{*}\bar{Q}Sy+\frac{1}{2}y^{*}S^{*}\bar{Q}Sy\\
    \qquad -\displaystyle\frac{1}{2}(P(t)x+\Sigma(t,m_{1})y)^{*}BR^{-1}B^{*}(P(t)x+\Sigma(t,m_{1})y)\\
    \qquad+(P(t)x+\Sigma(t,m_{1})y)^{*}(Ax+\bar{A}y).
\end{array}$$ Comparing coefficients, we obtain $$\left\{
\begin{array}{l}
\displaystyle\frac{d}{dt}P(t)+PA+A^{*}P-PBR^{-1}B^{*}P+Q+\bar{Q}=0,\label{eq:5.38}\\
P(T)=Q_{T}+\bar{Q}_{T}.
\end{array}
\right.$$ $$\left\{
\begin{array}{l}
\displaystyle\frac{d\Sigma}{dt}+\Sigma(A+\bar{A}m_{1}-BR^{-1}B^{*}P)+(A^{*}-PBR^{-1}B^{*})\Sigma-\Sigma BR^{-1}B^{*}\Sigma m_{1}-\bar{Q}S+P\bar{A}=0,\label{eq:5.39}\\
\Sigma(T,m_{1})=-\bar{Q}_{T}S_{T}.
\end{array}
\right.$$ $$\left\{
\begin{array}{l}
\displaystyle\frac{d\Gamma}{dt}+\Gamma(A+\bar{A}m_{1}-BR^{-1}B^{*}(P+\Sigma m_{1}))+(A+\bar{A}m_{1}-BR^{-1}B^{*}(P+\Sigma m_{1}))^{*}\Gamma\label{eq:5.40}\\
\qquad+S^{*}\bar{Q}S-\Sigma BR^{-1}B^{*}\Sigma+\Sigma\bar{A}+\bar{A}\,^{*}\Sigma=0,\\
\Gamma(T,m_{1})=S_{T}^{*}\bar{Q}_{T}S_{T}.
\end{array}
\right.$$ $$\left\{
\begin{array}{l}
\displaystyle\frac{d}{dt}\mu(t,m_{1})+\frac{1}{2}\text{tr}\, aP(t)+\frac{\beta^{2}}{2}\text{tr}P(t)+\frac{\beta^{2}}{2}\text{tr }\Gamma(t,m_{1})m_{1}^{2}+\beta^{2}\text{tr }\Sigma(t,m_{1})m_{1}=0,\label{eq:5.41}\\
\mu(T,m_{1})=0.
\end{array}
\right.$$

The function $\Gamma(t,m_{1})$ is no more the derivative of $\Sigma(t,m_{1})$ with respect to $m_{1}.$

\[prop2\] If we assume (\[eq:5.2\]) then the solution of the Master equation (\[eq:5.31\]) is given by formula (\[eq:5.33\]), with $P(t)$ solution of (\[eq:5.38\]), $\Sigma(t,m_{1})$ solution of (\[eq:5.39\]), $\Gamma(t,m_{1})$ solution of (\[eq:5.40\]) and $\mu(t,m_{1})$ solution of (\[eq:5.41\]).

SYSTEM OF STOCHASTIC HJB-FP EQUATIONS FOR MEAN FIELD GAMES 
-----------------------------------------------------------

We consider in the LQ case the system of stochastic HJB-FP equations for Mean Field games, namely (\[eq:4.3\]) and (\[eq:4.5\]). We first consider the FP equation, which is the same as for the Mean Field type control, namely (\[eq:5.25\]). With the notation of Section \[sub:Mean Field Type\], we have $$\left\{
\begin{array}{l}
dy=(A+\bar{A}-BR^{-1}B^{*}P(t))y(t)dt-BR^{-1}B^{*}r(t)dt+\beta db(t),\\
y(0)=\bar{x}_{0};
\end{array}
\right.$$ with $r(t)=\Sigma(t)y(t)$ and $\Sigma(t)=\Sigma(t,1)$. We obtain that $r(t)$ satisfies $$\left\{
\begin{array}{l}
-dr=(A^{*}-PBR^{-1}B^{*})rdt+(\bar{Q}S-P\bar{A})ydt-\beta\Sigma 