# Optimal stopping of oscillating Brownian motion

## TL;DR

This paper analyzes optimal stopping problems for oscillating Brownian motion with piecewise constant volatility, revealing conditions under which the continuation region becomes disconnected, and extends results to skew Brownian motion.

## Contribution

It provides the first characterization of the continuation region structure for optimal stopping of oscillating Brownian motion with a specific reward function.

## Key findings

- Continuation region is disconnected if and only if σ₁²<σ₂²<2σ₁².
- Results are extended to skew Brownian motion.
- Explicit conditions for the shape of the optimal stopping region.

## Abstract

We solve optimal stopping problems for an oscillating Brownian motion, i.e. a diffusion with positive piecewise constant volatility changing at the point $x=0$. Let $\sigma_1$ and $\sigma_2$ denote the volatilities on the negative and positive half-lines, respectively. Our main result is that continuation region of the optimal stopping problem with reward $((1+x)^+)^2$ is disconnected, if and only if $\sigma_1^2<\sigma_2^2<2\sigma_1^2$. Based on the fact that the skew Brownian motion in natural scale is an oscillating Brownian motion, the obtained results are translated into corresponding results for the skew Brownian motion.