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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
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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 

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