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

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.

31 pages, 8 figures. To appear in the Journal of Differential Equations

References in corpus (5)

• A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients
• Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations
• On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Ito processes
• Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance
• Maximum principles for boundary-degenerate second-order linear elliptic differential operators

Cited by in corpus (4)

• On the Heston Model with Stochastic Volatility: Analytic Solutions and Complete Markets
• Maximum principles for boundary-degenerate linear parabolic differential operators
• Degenerate-elliptic operators in mathematical finance and higher-order regularity for solutions to variational equations
• Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions