C^{1,1} regularity for degenerate elliptic obstacle problems

TL;DR

This paper proves optimal regularity results for solutions to obstacle problems involving the elliptic Heston operator, a degenerate elliptic PDE relevant in financial mathematics, using weighted functional spaces.

Contribution

It establishes the first $C^{1,1}$ regularity up to the boundary for solutions of the elliptic Heston obstacle problem with smooth obstacles.

Findings

Solutions are $C^{1,1}$ regular up to the boundary.

Weighted Sobolev and Holder spaces are effective tools.

Regularity results apply to financial models involving American options.

Abstract

The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to the obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset. With the aid of weighted Sobolev spaces and weighted Holder spaces, we establish the optimal $C^{1,1}$ regularity (up to the boundary of the half-plane) for solutions to obstacle problems for the elliptic Heston operator when the obstacle functions are sufficiently smooth.