Electronic Communications in Probability

Optimal stopping of oscillating Brownian motion

Ernesto Mordecki and Paavo Salminen

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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 $\sigma _{1}$ 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1568253712

Digital Object Identifier
doi:10.1214/19-ECP250

Mathematical Reviews number (MathSciNet)
MR4003124

Zentralblatt MATH identifier
1422.60134

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

Keywords
excessive function integral representation of excessive functions

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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