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Optimal stopping of oscillating Brownian motion. (English) Zbl 1422.60134
Summary: 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.

##### MSC:
 60J60 Diffusion processes 60J65 Brownian motion 62L15 Optimal stopping in statistics
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