Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T06:32:01.010Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal stopping rules for correlated random walks with a discount

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  14 July 2016

Pieter Allaart
Pieter Allaart*
Affiliation:
University of North Texas
*
∗ Postal address: Mathematics Department, University of North Texas, PO Box 311430, Denton, TX 76203-1430, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

Keywords

Correlated random walkstopping ruleoptimality principlediscount factormomentum

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G50: Sums of independent random variables; random walks
Secondary: 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 483 - 496
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaart, P. C., and Monticino, M. G. (2001). Optimal stopping rules for directionally reinforced processes. Adv. App. Prob. 33, 483–504.CrossRefGoogle Scholar
Böhm, W. (2000). The correlated random walk with boundaries: a combinatorial solution. J. Appl. Prob. 37, 470–479.CrossRefGoogle Scholar
Chen, A., and Renshaw, E. (1994). The general correlated random walk. J. Appl. Prob. 31, 869–884.CrossRefGoogle Scholar
Chow, Y. S., Robbins, H., and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston, MA.Google Scholar
Darling, D. A., Liggett, T., and Taylor, H. M. (1972). Optimal stopping for partial sums. Ann. Math. Statist. 43, 1363–1368.CrossRefGoogle Scholar
Dubins, L. E., and Teicher, H. (1967). Optimal stopping when the future is discounted. Ann. Math. Statist. 38, 601–605.CrossRefGoogle Scholar
Ferguson, T. S. (1976). Stopping a sum during a success run. Ann. Statist. 4, 252–264.CrossRefGoogle Scholar
Gillis, J. (1955). Correlated random walk. Proc. Camb. Phil. Soc. 51, 639–651.CrossRefGoogle Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. 4, 129–156.CrossRefGoogle Scholar
Henderson, R., and Renshaw, E. (1980). Spatial stochastic models and computer simulation applied to the study of tree root systems. Compstat 80, 389–395.Google Scholar
Iossif, G. (1986). Return probabilities for correlated random walks. J. Appl. Prob. 23, 201–207.CrossRefGoogle Scholar
Jain, G. C. (1971). Some results in a correlated random walk. Canad. Math. Bull. 14, 341–347.CrossRefGoogle Scholar
Jain, G. C. (1973). On the expected number of visits of a particle before absorption in a correlated random walk. Canad. Math. Bull. 16, 389–395.CrossRefGoogle Scholar
Mauldin, R. D., Monticino, M. and von Weizsäcker, H. (1996). Directionally reinforced random walks. Adv. Math. 117, 239–252.CrossRefGoogle Scholar
Mohan, C. (1955). The gambler's ruin problem with correlation. Biometrika 42, 486–493.CrossRefGoogle 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.CrossRefGoogle Scholar
Proudfoot, A. D., and Lampard, D. G. (1972). A random walk problem with correlation. J. Appl. Prob. 9, 436–440.CrossRefGoogle Scholar
Renshaw, E., and Henderson, R. (1981). The correlated random walk. J. Appl. Prob. 18, 403–414.CrossRefGoogle Scholar
Seth, A. (1963). The correlated unrestricted random walk. J. R. Statist. Soc. B 25, 394–400.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.CrossRefGoogle Scholar