I have a HJB equation that arises from a stochastic optimization problem.
$$u+a\partial_{x}V+b\partial_{y}V+\frac{\sigma^{2}}{2}\partial_{xx}V-\rho V=0$$
Where $V(x,y)$ is the unkown function and $u$, $a$, $b$ are possibly functions of $x$, $y$ and $V$, and $\sigma$ and $\rho$ are constants. Moreover, $x$ is can be any real number, while $n\in[0,1].$
I am trying to implement a upwind scheme numerically, approximating the derivatives $\partial_{x}V$ with forward differences when $a$ is positive and backward differences otherwise. Similarly, I am using forward differences for $\partial_{y}V$ iff $b>0$. For the second derivative, I am using central differences.
My question is: what should I do at the extreme of grid points? I tried forcing backward or forward differences, but the solution seems unstable and wrong, with some very small or very large derivatives near the boundaries of the grid.
I know that probably this equation has an unique viscosity solution (and that solution is the value function). But is there some finite difference scheme that is known to converge to the viscosity solution even when we do not have any boundary conditions?