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Leaving an interval in limited playing time

Published online by Cambridge University Press:  01 July 2016

David Heath  and
Robert P. Kertz
David Heath*
Affiliation:
Cornell University
Robert P. Kertz*
Affiliation:
Georgia Institute of Technology
*
∗Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA.
∗∗Postal address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
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Abstract

A player starts at x in (-G, G) and attempts to leave the interval in a limited playing time. In the discrete-time problem, G is a positive integer and the position is described by a random walk starting at integer x, with mean increments zero, and variance increment chosen by the player from [0, 1] at each integer playing time. In the continuous-time problem, the player's position is described by an Ito diffusion process with infinitesimal mean parameter zero and infinitesimal diffusion parameter chosen by the player from [0, 1] at each time instant of play. To maximize the probability of leaving the interval (–G, G) in a limited playing time, the player should play boldly by always choosing largest possible variance increment in the discrete-time setting and largest possible diffusion parameter in the continuous-time setting, until the player leaves the interval. In the discrete-time setting, this result affirms a conjecture of Spencer. In the continuous-time setting, the value function of play is identified.

Keywords

OPTIMAL STOCHASTIC CONTROLGAMBLING PROBLEMSBOLD PLAYEXCESSIVE FUNCTIONSEXIT TIMESBROWNIAN MOTIONRANDOM WALK
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 3 , September 1988 , pp. 635 - 645
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Supported in part by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University and by a grant from the AT&T Foundation.

Supported in part by National Science Foundation grants DMS-84–01604 and DMS-86–01153.

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