TITLE: Can we derive the PDE followed by a marginal transition probability density?
QUESTION [5 upvotes]: A pair of correlated stochastic processes follow the SDEs
\begin{align}
dX_t&=a(t,X_t)\,b(t,Y_t)\,dt+c(t,X_t)\,d(t,Y_t)\,dW_t, &&X_0=\bar{x}\\
dY_t&=f(t,Y_t)\,dt+g(t,Y_t)\,dZ_t, &&Y_0=\bar{y}
\end{align}
where $W_t$ and $Z_t$ are correlated Brownian motions with constant correlation $\rho$ and $a,b,c,d,f,g$ are smooth functions.
We know that the probability density $p(t,x,y)$ of the process reaching the state $X_t=x$, $Y_t=y$, given the initial condition at $t=0$ satisfies the forward Kolmogorov equation (also known as Fokker Planck equation):
$$
  p_t=-(abp)_x-(fp)_y+\frac{1}{2}(c^2d^2p)_{xx}+\frac{1}{2}(g^2p)_{yy}+\rho(cdgp)_{xy}
$$
Let $h(t,x)$ be the marginal probability distribution of the process $X_t$, i.e.
$$h(t,x)=\int_y p(t,x,y)dy$$
is it possible to write the partial differential equation in the variables $x$ and $t$ that $h(t,x)$ must satisfy?

REPLY [2 votes]: It is not possible to derive such a PDE as it doesn't exist in general because the process $X$ is not Markovian anymore. 
Note that in your case you could derive a PDE for the process $Y$ which is a one dimensional stochastic differential equation and so is Markovian if the solution exists.
Best regards.