Optimal stopping of oscillating Brownian motion

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 $((1+x)^{+})^{2}$ can be disconnected for some values of the discount rate when $2\sigma _{1}^{2}<\sigma _{2}^{2}$. 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.