I have hjb equation where $V=V(x,t)$ and $u=u(x,t)$$V_t + \sup(u) [A(x,u)V_x + B(x,u)V_{xx}]=0$ for $x$ in $[0,1]$ and $t$ in $[0,1]$I have been able to successfuly resolve it numerically having theese boundary conditions:$V(0,t)=0$ all t in [0,1]$V(1,t)=1$ all t in [0,1]$V(x,1)=x$ all x in [0,1]I used finite differences to compute $V_t$, $V_x$ and $V_{xx}$ and iterative process at each temporal step to find the optimal control $u^*$ recomputing the solution $V$ until differences among the steps are negligible.Now i would like to solve it on a infinite space boundary, so letting $x$ go from $-\inf$ to $+\inf$.I have the condition:$V(x,1)=max(x,0)$ for all $x$I thought to use:$V(x,t) \rightarrow 0$ for $x \rightarrow -\inf$ for all $t$ in $[0,1]$$V(x,t)/x \rightarrow 1$ for $x \rightarrow \inf$ for all $t$ in $[0,1]$how do you think is the best way to discretize these latter two conditions, if possible?Many thanks in advance, and i am sorry for the mistakes or imprecisions i can have made.
Finite Difference for Hamilton Jacobi Belman
finite difference;boundary conditions