I have something similar to a Fokker-Planck (aka forward Kolmogorov) equation. My equation is of the form $$\frac{\partial}{\partial t}f( x,t) = A(x,t)f(x,t)- \frac{\partial}{\partial x}[B(x,t) f(x,t)] +\frac{1}{2}\frac{\partial ^2}{\partial x^2 }[C(x,t) f(x,t)] \tag{1}\label{1}.$$
I am trying to obtain a corresponding stochastic differential equation. I know that if $A=0$, then \eqref{1} is the forward Kolmogorov equation corresponding to the diffusion SDE \eqref{2}, $$dX = B(X,t)dt + \sqrt{C(X,t)}dW. \tag{2}\label{2}$$
Is there a similar Ito SDE for $A\neq 0$?