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 σ1 and σ 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 σ 21 <σ22. 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.