Obstacle problems for nonlocal operators

TL;DR

This paper establishes existence, uniqueness, and regularity results for obstacle problems involving a broad class of nonlocal operators, including models used in finance, with solutions corresponding to American option prices.

Contribution

It introduces new conditions for regularity of viscosity solutions to nonlocal obstacle problems, covering non-stable-like operators with supercritical drift.

Findings

Proved existence and uniqueness of solutions.

Established Hölder and Lipschitz regularity.

Connected solutions to American option pricing models.

Abstract

We prove existence, uniqueness, and regularity of viscosity solutions to the stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. We give sufficient conditions on the coefficients of the operator to obtain H\"older and Lipschitz continuous solutions. The class of nonlocal operators that we consider include non-Gaussian asset price models widely used in mathematical finance, such as Variance Gamma Processes and Regular L\'evy Processes of Exponential type. In this context, the viscosity solutions that we analyze coincide with the prices of perpetual and finite expiry American options.