of the optimal state. This leads to the FP equation.

By analogy with what was done in the framework of mean field type control, the HJB equation becomes stochastic, and $m(t)$ becomes the conditional probability density. This motivates the model we develop in this section

THE MODEL 
----------

Adopt the notation of Section \[sub:PRELIMINARIES\]. We recall that the state equation is the solution of $$\left\{
\begin{split}
    dx & =g(x,m(t),v(x,t))dt+\sigma(x)dw+\beta db(t)\label{eq:4.1},\\
    x(0) & =x_{0};
\end{split}
\right.$$ and $\mathcal{B}^{t}$=$\sigma(b(s),s\leq t)$ , $\mathcal{F}$$^{t}$=$\sigma(x_{0},b(s),w(s),\: s\leq t)$. This time $m(t)$ is a given process adapted to $\mathcal{B}^{t}$ with values in $L^{2}(\mathbb{R}^{n}).$ We again consider feedback controls which are field processes adapted to the filtration $\mathcal{B}^{t}$.

We follow the theory developed by Shi-ge Peng [@SGP]. We define the function $$u(x,t)=\inf_{v(\cdot)}\:  \mathbb{E} ^{\mathcal{B}^{t}}[\int_{t}^{T}f(x(s),m(s),v(x(s),s))ds+h(x(T),m(T))]\label{eq:4.2}$$ and show that it satisfies a stochastic Hamilton-Jacobi-Bellman equation. Although, it is possible to proceed directly and formally from the definition of $u(x,t),$ to find out the stochastic HJB equation, the best is to postulate the equation and to use a verification argument. The equation is $$\left\{
\begin{array}{l}
    -\partial_{t}u+(\mathcal{A}u-\frac{1}{2}\beta^{2}\Delta u)dt  +\beta^{2}\text{div} B(x,t)dt=H(x,m(t),Du(x))dt+\beta B(x,t)db(t),\label{eq:4.3}\\
    u(x,T) =h(x,m(T)).
\end{array}
\right.$$

If we can solve (\[eq:4.3\]) for the pair $(u(x,t),\: B(x,t))$, $\mathcal{B}^{t}$-adapted field processes, then the verification argument 