Boundary-degenerate elliptic operators and Holder continuity for solutions to variational equations and inequalities

TL;DR

This paper establishes regularity results, including H"older continuity and bounds, for solutions to variational equations and inequalities involving the degenerate elliptic Heston operator, relevant in financial mathematics.

Contribution

It introduces weighted Sobolev space techniques to prove regularity and boundary behavior for solutions to the elliptic Heston operator, a degenerate-elliptic PDE.

Findings

Proved supremum bounds for solutions

Established a Harnack inequality near the boundary

Demonstrated H"older continuity up to the boundary

Abstract

The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for 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 whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. With the aid of weighted Sobolev spaces, we prove supremum bounds, a Harnack inequality, and H\"older continuity near the boundary for solutions to variational equations defined by the elliptic Heston operator, as well as H\"older continuity up to the boundary for solutions to variational inequalities defined by the elliptic Heston operator. In mathematical finance, solutions to obstacle problems for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.