Free Boundary Regularity in the Parabolic Fractional Obstacle Problem

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure-jump Levy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when s > 1/2, the free boundary is a C-1,C- graph in x and t near any regular free boundary point (x0,t0) is an element of partial derivative{u > phi}. Furthermore, we also prove that solutions u are C1+s in x and t near such points, with a precise expansion of the form u(x,t)-phi(x)=c(0)((x-x(0))center dot e+kappa(t-t(0)))(+)(1+s) + o(vertical bar x-x(0)vertical bar(1+s+alpha) + vertical bar t-t(0)vertical bar(1+s+alpha)), with c(0) > 0,e is an element of Sn-1, and a > 0. (c) 2018 Wiley Periodicals, Inc.