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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.

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