I'm slightly confused by the sketch proof of the Fokker Planck equation given to me in my lecture notes. I was hoping someone might be able to spread some light on the issue I'm having. It essentially boils down to an integration by parts problem I think.
This is the form I'm trying to derive:
\begin{equation} \frac{\partial}{\partial t}p(u,t;u_0,0)=-\frac{\partial}{\partial u}(u(t))p(u,t;u_0,0)+\frac{1}{2}\frac{\partial^2}{\partial u^2}((G(u(t))^2p(u,t;u_0,0)) \end{equation}
Sketch Proof
Given any smooth function $\phi$ we examine $\mathbf{E}^{u_0}[\phi(u(t))]:=\mathbb{E}[\phi(u(t))|u(0)=u_0]$. Therefore
\begin{equation} \mathbf{E}^{u_0}[\phi(u(t))]=\int_{\mathbb{R}}\phi(u(t))p(u,t;u_0,0)du \end{equation} Using the Ito formula and the fact that the expected value of Ito integrals are zero we have
\begin{equation} \mathbf{E}^{u_0}[\phi(u(t))]=\mathbf{E}^{u_0}[\phi(u(s))]+\mathbf{E}^{u_0}\bigg[\int_0^t \frac{\partial}{\partial u}\phi(u(s))f(u(s))+\frac{1}{2}\frac{\partial^2}{\partial u^2}\phi(u(s))(G(u(s))^2dt\bigg] \end{equation} Differentiate with respect to $t$ \begin{equation} \frac{d}{dt}\mathbf{E}^{u_0}[\phi(u(t))]=\mathbf{E}^{u_0}\bigg[\frac{\partial}{\partial u}\phi(u(t))f(u(t))+\frac{1}{2}\frac{\partial^2}{\partial u^2}\phi(u(t))(G(u(t))^2\bigg] \end{equation} And by the definition of expectation above \begin{equation} \int_{\mathbb{R}}\phi(u(t))\frac{\partial}{\partial t}p(u,t;u_0,0)du=\int_{\mathbb{R}}\bigg[\frac{\partial\phi}{\partial u}(u(t))f(u(t))+\frac{1}{2}\frac{\partial^2}{\partial u^2}\phi(u(t))(G(u(t))^2\bigg]p(u,t;u_0,0)du \end{equation} If we make assumptions on $p\rightarrow0$ as $u\rightarrow\infty$ we can use integration by parts once on $\frac{\partial \phi}{\partial u}fp$ and twice on $\frac{\partial^2 \phi}{\partial u^2}Gp$ to get \begin{equation} \int_{\mathbb{R}}\phi(u(t))\frac{\partial}{\partial t}p(u,t;u_0,0)du=\int_{\mathbb{R}}\phi(u)\bigg[-\frac{\partial}{\partial u}(u(t))p(u,t;u_0,0)+\frac{1}{2}\frac{\partial^2}{\partial u^2}((G(u(t))^2p(u,t;u_0,0))\bigg]du \end{equation} Since this holds for all smooth $\phi$ we must have \begin{equation} \frac{\partial}{\partial t}p(u,t;u_0,0)=-\frac{\partial}{\partial u}(u(t))p(u,t;u_0,0)+\frac{1}{2}\frac{\partial^2}{\partial u^2}((G(u(t))^2p(u,t;u_0,0)) \end{equation} The part I am struggling with is the where I have to use integration by parts. For some reason I just don't see how to get the next equation. If anyone could help I would really appreciate it. Many thanks.