# Finite Difference for Hamilton Jacobi Belman

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.

• Welcome to scicomp. This question fits the site very well. You can use latex commands to format your maths and to improve readability. – Jan Oct 25 '13 at 10:48
• Many thanks Jan for your welcome. I'll try to format it better next time :) – Isaac Asimov Oct 25 '13 at 14:37