Optimal stopping of oscillating Brownian motion.

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 $2sigma_1^2