Free boundary regularity close to initial state for parabolic obstacle problem

Abstract:

In this paper we study the behavior of the free boundary $\partial \{u>\psi \}$, arising in the following complementary problem: \begin{gather*} (Hu)(u-\psi )=0,\qquad u\geq \psi (x,t) \quad \mathrm {in}\ Q^+, Hu \leq 0, u(x,t) \geq \psi (x,t) \quad \mathrm {on}\ \partial _p Q^+. \end{gather*} 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 $\textbf {R}^{n+1}$, 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.