Hypoelliptic Regularization in the Obstacle Problem for the Kolmogorov Operator

TL;DR

This paper improves the regularity results for solutions to the obstacle problem associated with the Kolmogorov operator, leading to new insights into free boundary regularity using advanced techniques.

Contribution

It advances the regularity theory for the obstacle problem of the Kolmogorov operator and establishes the first free boundary regularity result under standard conditions.

Findings

Enhanced regularity from $C^{0,1}_t \cap C^{0,2/3}_x \cap C^{1,1}_v$ to $C^{0,1}_{t,x} \cap C^{1,1}_v$

First free boundary regularity result showing $C^{0,1/2}_{t,x} \cap C^{0,1}_v$ regular surface

Introduction of a new monotonicity formula and commutator estimate

Abstract

We study the obstacle problem associated with the Kolmogorov operator $\Delta_v - \partial_t - v\cdot\nabla_x$, which arises from the theory of optimal control in Asian-American options pricing models. Our first main contribution is to improve the known regularity of solutions, from $C^{0,1}_t \cap C^{0,2/3}_x \cap C^{1,1}_v$ to $C^{0,1}_{t,x} \cap C^{1,1}_v$. The previous result in the literature, which has been called optimal, corresponds to $C^{1,1}$ regularity with respect to the Kolmogorov distance. This is the expected regularity for solutions to obstacle problems. Our unexpected improvement of regularity in the $x$ variable is obtained using Bernstein's technique and an approach drawing on ideas from Evans-Krylov theory. We then use this improvement in regularity of the solution to prove the first known free boundary regularity result. We show that under a standard thickness condition, the free boundary is a $C^{0,1/2}_{t,x} \cap C^{0,1}_v$ regular surface. This result constitutes the first step in the program of free boundary regularity. Critically, our arguments rely on a new monotonicity formula and a commutator estimate that are only made possible by the solution's enhanced regularity in $x$.