Electronic Communications in Probability

Optimal stopping of oscillating Brownian motion

Ernesto Mordecki and Paavo Salminen

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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.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 50, 12 pp.

Received: 4 March 2019
Accepted: 19 June 2019
First available in Project Euclid: 12 September 2019

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65] 62L15: Optimal stopping [See also 60G40, 91A60]

excessive function integral representation of excessive functions

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Mordecki, Ernesto; Salminen, Paavo. Optimal stopping of oscillating Brownian motion. Electron. Commun. Probab. 24 (2019), paper no. 50, 12 pp. doi:10.1214/19-ECP250. https://projecteuclid.org/euclid.ecp/1568253712

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