I want to solve computationally for a function $V: \mathbb{R^2} \rightarrow \mathbb{R}$ which is known to be convex. When V is $C^2$ I know the function satisfies, $$\rho V(x,z) = \max_{x} f(x) + V_x(x,z) \dot x + \mu V_z(x,z) +\sigma 1/2 V_{zz}(x,z)$$ where $\rho, \mu, \sigma$ are arbitrary constants. f is a convex function and $\dot x$ is the derivative of $x$ with respect to time, which follows a rule given by my problem.
The formula above is obtained using Ito's formula. And when V is $C^2$ all I need to do is solve the above differential equation using a finite difference method.
However in general V might have kinks, and I can derive the corresponding differential equation using the generalized Ito rule for convex functions. Now, it seems to me that the algorithm to solve this using a finite different method is unchanged.
More specifically, in the analytical formula, at the kinks I get "correction terms" $$L(a) [V_x^+ - V_x^-]$$ where $V_x^+$, $V_x^-$ are right and left derivatives and $L$ is the local time at the kink $a$.
It seems to me that this "correction terms" are taken into account when the second derivatives are being approximated using finite differences.
Is that correct? My reasoning is all based on heuristics and I lack a formal proof showing that the finite difference algorithm would converge to the answer.