Hypoelliptic Regularization in the Obstacle Problem for the Kolmogorov Operator

Abstract We study the obstacle problem associated with the Kolmogorov operator $$\mathcal {L}{:}{=}\Delta _v $$ L : = Δ v $$- \partial _t - v\cdot \nabla _x$$ - ∂ t - v · ∇ x , which arises from the theory of optimal control in Asian-American options pricing models. This problem presents a significant departure from the elliptic and parabolic obstacle problems due to the highly degenerate hypoelliptic nature and the non-commutative Galilean group structure underlying the operator $$\mathcal {L}$$ L . First, we improve the known regularity of solutions. The existing interior estimate in the literature is from [14], where it is proven that solutions are $$C^{1,1}$$ C 1 , 1 with respect to the kinetic distance and therefore $$C^{0,2/3}_x$$ C x 0 , 2 / 3 Hölder regular in the spatial variable x . This result, which is claimed to be optimal, is the typical regularity obtained in obstacle problems, and is consistent with the typically expected gain of 2/3 derivatives in x from the theory of hypoellipticity for the operator $$\mathcal {L}$$ L . Our first result is rather unexpected then, for we manage to prove an even stronger hypoelliptic regularization effect: solutions enjoy a Lipschitz $$C^{0,1}_x$$ C x 0 , 1 interior estimate in the spatial variable x . We prove $$C^{0,1}_{t,x} \cap C^{1,1}_v$$ C t , x 0 , 1 ∩ C v 1 , 1 interior estimates following an approach which is based on Bernstein’s technique and draws on ideas from Evans-Kyrlov theory, and our method seems to be quite robust. 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 differentiable with respect to the kinetic distance and is a $$C^{0,1/2}_{t,x} \cap C^{0,1}_v$$ C t , x 0 , 1 / 2 ∩ C v 0 , 1 regular surface. This result con