Free Boundary Regularity Close to Initial State for Parabolic Obstacle Problem

In this paper we study the behavior of the free boundary ∂{u > ψ}, arising in the following complementary problem: (Hu)(u− ψ) = 0, u ≥ ψ(x, t) in Q, Hu ≤ 0, u(x, t) ≥ ψ(x, t) on ∂pQ. Here ∂p denotes the parabolic boundary, H is a parabolic operator with certain properties, Q+ is the upper half of the unit cylinder in R, and the equation is satisfied in the viscosity sense. The obstacle ψ is assumed to be continuous (with a certain smoothness at {x1 = 0, t = 0}), and coincides with the boundary data u(x, 0) = ψ(x, 0) at time zero. We also discuss applications in financial markets.