Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T18:43:57.013Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal double stopping of a Brownian bridge

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  21 March 2016

Erik J. Baurdoux ,
Nan Chen ,
Budhi A. Surya  and
Kazutoshi Yamazaki
Erik J. Baurdoux*
Affiliation:
London School of Economics
Nan Chen*
Affiliation:
The Chinese University of Hong Kong
Budhi A. Surya*
Affiliation:
Victoria University of Wellington and Bandung Institute of Technology
Kazutoshi Yamazaki*
Affiliation:
Kansai University
*
∗ Postal address: Department of Statistics, London School of Economics, London WC2A 2AE, UK.
∗∗ Postal address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong.
∗∗∗ Postal address: School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand.
∗∗∗∗ Postal address: Department of Mathematics, Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho Suita, Osaka, 564-8680, Japan. Email address: kyamazak@kansai-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study optimal double stopping problems driven by a Brownian bridge. The objective is to maximize the expected spread between the payoffs achieved at the two stopping times. We study several cases where the solutions can be solved explicitly by strategies of a threshold type.

Keywords

Brownian bridgeoptimal double stoppingbuying-selling strategy

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 1212 - 1234
Copyright
Copyright © Applied Probability Trust 2015 

References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.Google Scholar
Avellaneda, M. and Lipkin, M. D. (2003). A market-induced mechanism for stock pinning. Quant. Finance 3, 417–425.CrossRefGoogle Scholar
Ekström, E. and Wanntorp, H. (2009). Optimal stopping of a Brownian bridge. J. Appl. Prob. 46, 170–180.Google Scholar
Ekström, E., Lindberg, C. and Tysk, J. (2011). Optimal liquidation of a pairs trade. In Advanced Mathematical Methods for Finance, Springer, Heidelberg, pp. 247–255.Google Scholar
Gallego, G. and van Ryzin, G. (1994). Optimal dynamic pricing of inventories with stochastic demand over finite horizons. Manag. Sci. 40, 999–1020.CrossRefGoogle Scholar
Ivashko, A. A. (2014). Gain maximization problem in the urn scheme. In Transactions of Karelian Research Centre of Russian Academy of Science, No 4. Mathematical Modeling and Information Technologies, pp. 62–66.Google Scholar
Leung, T. and Li, X. (2015). Optimal mean reversion trading with transaction cost and stop-loss exit. Internat. J. Theoret. Appl. Finance 18, 1550020.Google Scholar
Lin, T. C. W. (2013). The new investor. UCLA Law Rev. 60, 678–735.Google Scholar
Mazalov, V. V. and Tamaki, M. (2007). Duration problem on trajectories. Stochastics 79, 211–218.Google Scholar
Peskir, G. (2005). A change-of-variable formula with local time on curves. J. Theoret. Prob. 18, 499–535.CrossRefGoogle Scholar
Shepp, L. A. (1969). Explicit solutions to some problems of optimal stopping. Ann. Math. Statist. 40, 993–1010.Google Scholar
Sofronov, G., Keith, J. M. and Kroese, D. P. (2006). An optimal sequential procedure for a buying-selling problem with independent observations. J. Appl. Prob. 43, 454–462.CrossRefGoogle Scholar
Song, Q. S., Yin, G. and Zhang, Q. (2009). Stochastic optimization methods for buying-low-and-selling-high strategies. Stoch. Anal. Appl. 27, 523–542.Google Scholar
Tamaki, M. (2001). Optimal stopping on trajectories and the ballot problem. J. Appl. Prob. 38, 946–959.Google Scholar