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.
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