American option model and negative Fichera function on degenerate boundary

We study American put option with stochastic volatility whose value function is associated with a 2-dimensional parabolic variational inequality with degenerate boundaries. Given the Fichera function on the boundary, we first analyze the existences of the strong solution and the properties of the 2-dimensional manifold for the free boundary. Thanks to the regularity result of the underlying PDE, we can also provide the uniqueness of the solution by the argument of the verification theorem together with the generalized Itos formula even though the solution may not be second order differentiable in the space variable across the free boundary.