Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T01:47:01.110Z Has data issue: false hasContentIssue false
  • English
  • Français

An unexpected connection between branching processes and optimal stopping

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

David Assaf ,
Larry Goldstein  and
Ester Samuel-Cahn
David Assaf*
Affiliation:
Hebrew University of Jerusalem
Larry Goldstein*
Affiliation:
University of Southern California
Ester Samuel-Cahn*
Affiliation:
Hebrew University of Jerusalem
*
∗Postal address: Department of Statistics, Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel
∗∗Postal address: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
∗Postal address: Department of Statistics, Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Zn with offspring distribution Y, there exists a random variable X such that the probability P(Zn = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X1,…, Xn, where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

Keywords

Optimal stopping problemGalton-Watson processextinction probabilityvarying environmentinhomogeneous Galton-Watson processprophet inequalities

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 613 - 626
Copyright
Copyright © Applied Probability Trust 2000 

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

Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Chow, Y., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston.Google Scholar
D'Souza, J. C. (1995). The extinction time of the inhomogeneous branching process. In Branching Processes, ed. Heyde, C. C. (Lecture Notes in Statist. 99). Springer, Berlin, pp. 106–117.CrossRefGoogle Scholar
De Haan, L. (1976). Sample extremes: an elementary introduction. Statist. Neerlandica 30, 161–172.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Hill, T. P., and Kertz, R. P. (1981). Ratio comparisons of supremum and stop rule expectations. Z. Wahrscheinlichkeitsth. 56, 283–285.Google Scholar
Hill, T. P., and Kertz, R. P. (1982). Comparisons of stop rule and supremum expectations of i.i.d. random variables. Ann. Prob. 10, 336–345.Google Scholar
Jagers, P. (1974). Galton–Watson processes in varying environments. J. Appl. Prob. 11, 174–178.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Jirina, M. (1976). Extinction of non-homogeneous Galton–Watson processes. J. Appl. Prob. 13, 132–137.CrossRefGoogle Scholar
Karlin, S., and Taylor, H. M. (1975). A First Course in Stochastic Processes. Second edn., Academic Press, New York.Google Scholar
Keiding, N., and Nielsen, J. E. (1975). Branching processes with varying and geometric offspring distribution. J. Appl. Prob. 12, 135–141.CrossRefGoogle Scholar
Kennedy, D. P., and Kertz, R. P. (1991). The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables. Ann. Prob. 9, 329–341.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Slack, R. S. (1968). A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.CrossRefGoogle Scholar