The proof of the concavity of the value function by the methods of PDE and stochastic analysis are given separately.

We prove that the equation is degenerate in the left boundary but non-degenerate interior.

We give high regularity of the solution (value function), we prove it is 3 times differentiable.

We prove that the free boundary h(t)$h(t)$ is a differentiable curve and show its upper and lower bounds and starting point h(0)=0$h(0)=0$.