is as follows: for any $\mathcal{B}^{t}$-adapted field process $v(x,t)$, we write $$H(x,m(t),Du(x,t))\leq f(x,m(t),v(x,t))+Du(x,t)g(x,m(t),v(x,t)).$$

Consider the process $x(t)$, the solution of the state equation (\[eq:4.1\]), we compute the Ito differential $d\, u(x(t),t)$ by using a generalized Itô’s formula due to Kunita, which gives $$\begin{array}{l}
    du(x(t),t)=(Du g(x(t),m(t),v(x(t),t))+\displaystyle\frac{1}{2}\text{tr}(a(x(t))D^{2}u)+\frac{\beta^{2}}{2}\Delta u)dt\\
    \qquad\qquad\qquad\qquad+Du(\sigma(x(t))dw+\beta db(t))+\partial_{t}u(x(t),t)-\beta^{2}\text{div}(B(x(t),t))dt.
\end{array}$$ Hence we have the inequality, by using (\[eq:4.3\]), $$\begin{array}{l}
    h(x(T),m(T))+\displaystyle\int_{t}^{T}f(x(s),m(s),v(x(s),s))ds\\
    \qquad\qquad\geq u(x,t)+\displaystyle\int_{t}^{T}Du(x(s))(\sigma(x(s))dw(s)+\beta db(s))-\beta\int_{t}^{T}B(x(s))db(s),
\end{array}$$ from which we get $$u(x,t)\leq  \mathbb{E} ^{\mathcal{B}^{t}}[\int_{t}^{T}f(x(s),m(s),v(x(s),s))ds+h(x(T),m(T))],$$ and a similar development used for the optimal feedback yields the equality, hence the property (\[eq:4.2\]) follows.

Next, consider the optimal feedback $\hat{v}(x,m(t),Du(x,t))$ and impose the fixed point property that $m(t)$ is conditional probability density of the optimal state, we get the stochastic FP equation $$\left\{
\begin{array}{l}
\partial_{t}m+(\mathcal{A}^{*}m-\frac{1}{2}\beta^{2}\Delta m+\text{div}(G(x,m,Du)m))dt+\beta Dmdb(t)=0,\label{eq:4.5}\\
m(x,0)=m_{0}(x).
\end{array}
\right.$$

We thus have obtained the pair of HJB-FP equations for the stochastic mean field game problem, (\[eq:4.3\]) and (\[eq:4.5\]).

THE MASTER EQUATION
-------------------

We shall derive the Master equation by writing $u(x,t)=U(x,m(t),t)$. We must have, by using (\[eq:4.5\]), $$\begin{array}{l}
    \partial_{t}u=\bigg[\displaystyle\frac{\partial U}{\partial t}-\int_{\mathbb{R}^{n}}\frac{\partial U(x,m(t),t)}{\partial m}(\xi)(\mathcal{A}^{*}m(\xi)-\frac{1}{2}\beta^{2}\Delta m(\xi)+\text{div }(G(\xi,m(t),DU(\xi))m(\xi)))d\xi\\
    \qquad\qquad\qquad\qquad+\displaystyle\frac{1}{2}\beta^{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\partial^{2}U(x,m(t),t)}{\partial m^{2}}(\xi,\eta)Dm(\xi)Dm(\eta)d\xi d\eta\bigg]dt\\
    \qquad\qquad-\beta\displaystyle\int_{\mathbb{R}^{n}}\frac{\partial U(x,m(t),t)}{\partial m}(\xi)Dm(\xi)d\xi db(t).
\end{array}$$

Comparing terms with (\[eq:4.3\]), and letting $$B(x,t)=\int_{\mathbb{R}^{n}}\frac{\partial U(x,m(t),t)}{\partial m}(\xi)Dm(\xi)d\xi,$$ we have the master equation $$\label{eq:4.7}
\left\{
\begin{array}{l}
    -\displaystyle\frac{\partial U}{\partial t}+\mathcal{A}U-\frac{1}{2}\beta^{2}\Delta U+\int_{\mathbb{R}^{n}}\frac{\partial U(x,m,t)}{\partial m}(\xi)\Big(\mathcal{A}^{*}m(\xi)-\frac{1}{2}\beta^{2}\Delta 