I'm struggling to proof the Fokker-Planck equation.

Let $b:[0,T]\times \mathbb{R}^N\to\mathbb{R}^N$ and $\sigma:[0,T]\times \mathbb{R}^N\to\mathbb{R}^{N\times d}$ two measurable functions.

Let $X=\begin{Bmatrix} X_t \end{Bmatrix}_{t\in[0,T]}$ a multivariate Ito's process with dynamic $dX_t:=b(t,X_t)dt+\sigma(t,X_t)dW_t$ and $b(t,X_t)=N\times 1$ , $\sigma(t,X_t)=N\times d$.

I know for hypothesis that $\forall(t,x)\in S_t\subseteq \mathbb{R}^N,X_t^{t,x}$ is solution for $dX_t$ such that $X_t^{t,x}=x$ in $(\Omega,F,\begin{Bmatrix} F_t \end{Bmatrix}_{t\geq 0},\mathbb{P})$. First problem: what's the meaning of this condition? $t,x$ on apex means that the solution $X_t$ for $dX_t$ depends on the time and the process itself?

Then, let $f(X_t)\in C^2(\mathbb{R}^N)$ a continuous and 2-derivable function (with derivatives defined on $\mathbb{R}^N$) with multivariate Ito's dynamic $df(X_t)=\bigtriangledown f(X_t)\sigma(t,X_t)dW_t+\frac{1}{2}[\sum_{i,j=1}^{N}c_{i,j}(t,X_t)\frac{\partial^2f(X_t)}{\partial X_t^i \partial X_t^j}+\sum_{j=1}^{N}b_j(t,X_t)\frac{\partial f(X_t)}{\partial X_t^j}]dt$.

Since $f$ depends only on the process and not on the time, we have not the spatial derivative of the time. Furthermore, $c_{i,j}$ is the i,j-th component of the quadratic and symmetric matrix that we obtain by multiplying $\sigma(t,X_t)=N\times d$ with the transposed $\sigma^*(t,X_t)=d\times N$.

Let also $A_t:=\frac{1}{2}[\sum_{i,j=1}^{N}c_{i,j}(t,X_t)\frac{\partial^2}{\partial X_t^i \partial X_t^j}+\sum_{j=1}^{N}b_j(t,X_t)\frac{\partial}{\partial X_t^j}]$ the characteristic operator of SDE $dX_t$. So $df(X_t)=A_tf(X_t)dt+\bigtriangledown f(X_t)\sigma(t,X_t)dW_t$.

So, we have the PdC $\left\{\begin{matrix} Au-au+\frac{\partial u}{\partial t}=f\\ u(T,\cdot)=\phi \end{matrix}\right.$ with $f,a,\phi$ known functions.

Good. Now, Feynman-Kac THM says that $u(t,x)=\mathbb{E}[e^{-\int_{t}^{T}a(s,X_s)ds}\phi(X_T)-\int_{t}^{T}e^{-\int_{s}^{t}a(r,X_r)dr}f(s,X_s)ds]$ is solution of PdC under the regularity hp. for $b$ and $\sigma$. Second problem: i don't know why.

Then, let $F:=A_t+\frac{\partial}{\partial t}$ with solution $P$. If $\phi$ is continuous and derivable it's shown that $u(t,x)=\int_{\mathbb{R}^N}\phi (y)P(t,y|T,x)dy$. Third problem: i don't know why.

By placing $a=f=0$ we obtain that $u(t,x)=\mathbb{E}[\phi(X_T^{t,x})]$ (with $t,x$ that I still don't understand).

So, it's shown (Fourth problem: i don't know why) that if exists a solution for $F:=A_t+\frac{\partial}/{\partial t}$, that solution is the transition density of $dX_t$.

In conclusion we obtain (Fifth problem: i don't know why) that the F-P equation is $\frac{\partial P}{\partial t}=\frac{1}{2}\sum_{i,j=1}^{N}\frac{\partial}{\partial X_i \partial X_j}(c_{i,j}P)-\sum_{j=1}^{N}\frac{\partial}{\partial X_j}(b_jP)$.

I apologize for all this unclear points but my text is too dispersed.

I really hope that some of you can help me. Thanks in advance!


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