Free boundary regularity close to initial state for parabolic obstacle problem

In this paper we study the behavior of the free boundary $\partial \{u>\psi \}$, arising in the following complementary problem:

$(Hu)(u-\psi)=0,\qquad u\geq \psi (x,t) \quad \hbox{in } Q^+,$
$Hu \leq 0,$
$u(x,t) \geq \psi (x,t) \quad \hbox{on } \partial_p Q^+.$

Here $\partial_p$ denotes the parabolic boundary, $H$ is a parabolic operator with certain properties, $Q^+$ is the upper half of the unit cylinder in ${\bf R}^{n+1}$, and the equation is satisfied in the viscosity sense. The obstacle $\psi$ is assumed to be continuous (with a certain smoothness at $\{x_1=0$, $t=0\}$), and coincides with the boundary data $u(x,0)=\psi (x,0)$ at time zero. We also discuss applications in financial markets.