# Boundary Conditions for Fokker Plank (Kolmogorov Forward Equation)

I have a stochastic process given by $$dX(t)=X(1-X)dtv+X(1-X)\sigma dW(t)$$ Once X reaches $$0$$ or $$1$$, the process stays there for ever. (the drift and diffusion terms goes to zero). I would have liked to derive the expected value for this process analytically. But as far as I can see, this is impossible. Since this is a non linear SDE, one cannot derive the the laws of motion of the moments in a way that's useful. So my next step was to write down the Fokker plank equation associated with this process. That is $$\frac{\partial \rho(x,t)}{\partial t}=-\frac{\partial}{\partial x}[\underbrace{\mu(x)\rho-\frac{\partial}{\partial x}(D(x)\rho)]}_{J(x,t)}$$ where $$\mu(x)=vx(1-x)$$ and $$D(x)=\frac{(x(1-x)\sigma)^2}{2}$$ and $$J$$ represents the probability current. I would like to solve this numerically. But i am unable to decide which boundary conditions would be appropriate. The most natural to me seems to mixed-type boundary conditions. We would expect the probability current to be zero at the edges of $$0$$ and $$1$$. But to me , it does not quite capture the fact that once the state has reached the boundary, it stays there forever.