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Optimal stopping rules for directionally reinforced processes

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Pieter Allaart  and
Michael Monticino
Pieter Allaart*
Affiliation:
University of North Texas
Michael Monticino*
Affiliation:
University of North Texas
*
∗ Postal address: Mathematics Department, University of North Texas, Denton, TX 76203-1430, USA.
∗ Postal address: Mathematics Department, University of North Texas, Denton, TX 76203-1430, USA.
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Abstract

This paper analyzes optimal single and multiple stopping rules for a class of correlated random walks that provides an elementary model for processes exhibiting momentum or directional reinforcement behavior. Explicit descriptions of optimal stopping rules are given in several interesting special cases with and without transaction costs. Numerical examples are presented comparing optimal strategies to simpler buy and hold strategies.

Keywords

correlated random walkdirectionally reinforced random walkmultiple stoppingbuy/sell strategies

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 483 - 504
Copyright
Copyright © Applied Probability Trust 2001 

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References

Böhm, W., (2000). The correlated random walk with boundaries: a combinatorial solution. J. Appl. Prob. 37, 470–479.Google Scholar
Bruss, F. T. and Paindaveine, D. (2000). Selecting a sequence of last successes in independent trials. J. Appl. Prob. 37, 389–399.CrossRefGoogle Scholar
Chen, A. Y. and Renshaw, E. (1994). The general correlated random walk. J. Appl. Prob. 31, 869–884.Google Scholar
Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.Google Scholar
Gilbert, J. P. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.CrossRefGoogle Scholar
Gillis, J. (1955). Correlated random walk. Proc. Camb. Phil. Soc. 51, 639–651.Google Scholar
Haggstrom, G. W. (1966). Optimal stopping and experimental design. Ann. Math. Statist. 37, 7–29.CrossRefGoogle Scholar
Horváth, L. and Shao, O. M. (1998). Limit distributions of directionally reinforced random walks. Adv. Math. 134, 367–383.Google Scholar
Kao, M. Y. and Tate, S. R. (1999). On-line difference maximization. SIAM J. Discrete Math. 12, 78–90.Google Scholar
Majumdar, A. A. K. (1983). Optimal stopping based on success runs for an urn with three types of balls. Ganit 3, 1–30.Google Scholar
Majumdar, A. A. K. (1989). Optimal stopping based on success or failure run. Indian J. Math. 31, 175–192.Google Scholar
Majumdar, A. A. K. and Sakaguchi, M. (1983). Optimal stopping for the urn problem with random termination. Math. Japon. 28, 271–285.Google Scholar
Mauldin, R. D., Monticino, M. G. and Von Weizsäcker, H. (1996). Directionally reinforced random walks. Adv. Math. 117, 239–252.Google Scholar
Mohan, C. (1955). The gamblerés ruin problem with correlation. Biometrika 42, 486–493.CrossRefGoogle Scholar
Mukherjea, A. and Steele, D. (1986). Conditional expected durations of play given the ultimate outcome for a correlated random walk. Statist. Prob. Lett. 4, 237–243.Google Scholar
Mukherjea, A. and Steele, D. (1987). Occupation probability of a correlated random walk and a correlated ruin problem. Statist. Prob. Lett. 5, 105–111.Google Scholar
Ross, S. M. (1975). A note on optimal stopping for success runs. Ann. Statist. 3, 793–795.Google Scholar
Stadje, W. (1985). On multiple stopping rules. Optimization 16, 401–418.Google Scholar
Starr, N. (1972). How to win a war if you must: optimal stopping based on success runs. Ann. Math. Statist. 43, 1884–1893.Google Scholar
Zhang, Y. L. (1992). Some problems on a one-dimensional correlated random walk with various types of barrier. J. Appl. Prob. 29, 196–201.Google Scholar