We have solved M. problem?

The problem becomes quite interesting once I apply the viscosity solution theory.
Actually the second derivative Y_{xx} might not be always positive. (I made a terrible assumption last time by assuming Y to be strictly concave.) Hence, the value function Y(x) might be convex for small x and concave for larger x. It is very nice, since when it's convex, the value function satisfies the Black-Scholes-Merton equation. We should be fascinated about our result.
Cannot wait to discuss it with "Big Boss".