Hypoelliptic Regularization in the Obstacle Problem for the Kolmogorov Operator

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 $x$.