# On Lipschitz continuous optimal stopping boundaries

## TL;DR

This paper provides a probabilistic proof of the local Lipschitz continuity of optimal stopping boundaries for certain stochastic processes, extending previous results by removing the need for PDE-based methods and uniform ellipticity assumptions.

## Contribution

It introduces a novel probabilistic approach to establish Lipschitz continuity of optimal stopping boundaries, applicable to non-uniformly elliptic diffusions.

## Key findings

- Proves Lipschitz continuity using stochastic calculus only.
- Extends results to non-uniformly elliptic diffusions.
- Provides a new probabilistic framework for optimal stopping boundaries.

## Abstract

We obtain a probabilistic proof of the local Lipschitz continuity for the optimal stopping boundary of a class of problems with state space $[0,T]\times\mathbb{R}^d$, $d\ge 1$. To the best of our knowledge this is the only existing proof that relies exclusively upon stochastic calculus, all the other proofs making use of PDE techniques and integral equations. Thanks to our approach we obtain our result for a class of diffusions whose associated second order differential operator is not necessarily uniformly elliptic. The latter condition is normally assumed in the related PDE literature.