| Abstract: |
| We study the obstacle problem associated with the Kolmogorov operator $\mcL \coloneqq \Delta_v - \partial_t - v\cdot\nabla_x$, 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 $\mcL$.
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 corresponds to $C^{1,1}$ regularity with respect to the Kolmogorov distance, which is the expected regularity for obstacle problems. Our method uses Bernstein`s technique and draws on ideas from Evans-Krylov theory.
We then use this improvement in regularity of the solution to prove the first 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. Our arguments rely on a new monotonicity formula and a commutator estimate that are only made possible by the solution`s enhanced regularity in $x$. |
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