problem is well-posed.

STOCHASTIC MEAN FIELD TYPE CONTROL
==================================

\[sub:PRELIMINARIES\]PRELIMINARIES
----------------------------------

If we look at the formulation (\[eq:2.1\]), (\[eq:2.2\]) of the mean field type control problem, it is a deterministic problem, although at the origin it was a stochastic one, see (\[eq:1.1\]), (\[eq:1.2\]). We now consider a stochastic version of (\[eq:2.1\]), (\[eq:2.2\]) or a doubly stochastic version of (\[eq:1.1\]), (\[eq:1.2\]). Let us begin with this one. Assume that there is a second standard Wiener process $b(t)$ with values in $\mathbb{R}^{n}$; $b(t)$ and $w(t)$ are independent which are also independent of $x_{0}$. We set $\mathcal{B}^{t}$=$\sigma(b(s):s\leq t)$ and $\mathcal{F}$$^{t}$=$\sigma(x_{0},b(s),w(s):\: s\leq t)$ . The control $v(x,t)$ at time $t$ is a feedback, but not deterministic, i.e. the functional form of $v$ is $\mathcal{B}^{t}$ adapted. We consider the stochastic McKean-Vlasov equation $$\label{eq:3.1}
\left\{
\begin{split}
dx&=g(x,m_{v(\cdot)}(t),v(x,t))dt+\sigma(x)dw+\beta db(t),\\
x(0)&=x_{0},\\
\end{split}
\right.$$ in which $m_{v(\cdot)}(t)$ represents the conditional probability density of $x(t)$, given the $\sigma$-algebra $\mathcal{B}^{t}$ . The stochastic mean field type control problem aims at minimizing the objective functional $$J(v(\cdot))= \mathbb{E} \bigg[\int_{0}^{T}f(x(t),m_{v(\cdot)}(t),v(x(t),t))dt+h(x(T),m_{v(\cdot)}(T))\bigg].\label{eq:3.2}$$

CONDITIONAL PROBABILITY
-----------------------

Let $y(t)=x(t)- \beta b(t)$, then the process $y(t)$ satisfies the equation $$\label{eq:3.1-1}
\left\{
\begin{split}
&dy=g(y(t)+\beta b(t),m_{v(\cdot)}(t),v(y(t)+\beta b(t),t))dt+\sigma(y(t)+\beta b(t))dw,\\
&y(0)=x_{0}.\\
\end{split}
\right.$$ If we fix $b(s),s\leq t$, then the conditional probability of 