Free boundary regularity in the parabolic fractional obstacle problem

TL;DR

This paper investigates the regularity of the free boundary in the parabolic fractional obstacle problem, showing that for s > 1/2, the free boundary is a smooth graph and solutions exhibit precise regularity near regular points.

Contribution

It establishes the $C^{1,eta}$ regularity of the free boundary and detailed expansion of solutions near regular points for the parabolic fractional obstacle problem when s > 1/2.

Findings

Free boundary is a $C^{1,eta}$ graph near regular points.

Solutions are $C^{1+s}$ regular near such points.

Precise asymptotic expansion of solutions near regular points.

Abstract

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump L\'evy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when $s>\frac12$, the free boundary is a $C^{1,\alpha}$ graph in $x$ and $t$ near any regular free boundary point $(x_0,t_0)\in \partial\{u>\varphi\}$. Furthermore, we also prove that solutions $u$ are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion of the form \[u(x,t)-\varphi(x)=c_0\bigl((x-x_0)\cdot e+a(t-t_0)\bigr)_+^{1+s}+o\bigl(|x-x_0|^{1+s+\alpha}+ |t-t_0|^{1+s+\alpha}\bigr),\] with $c_0>0$, $e\in \mathbb{S}^{n-1}$, and $a>0$.