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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.

References

[1] Dubins, L. E. and Savage, L. J. (1965), (1976) How to Gamble If You Must: Inequalities for Stochastic Processes. Dover, New York.Google Scholar
[2] Durrett, R. (1984) Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, California.Google Scholar
[3] Freedman, D. (1971) Brownian Motion and Diffusion. Holden-Day, San Francisco.Google Scholar
[4] Heath, D. and Kertz, R. P. (1986) Leaving an interval in limited playing time. Technical Report, School of Mathematics, Georgia Institute of Technology, Atlanta.Google Scholar
[5] Heath, D., Orey, S., Pestien, V., and Sudderth, W. (1987) Minimizing or maximizing the expected time to reach zero. SIAM J. Control Optim. 25, 195–205.CrossRefGoogle Scholar
[6] Heath, D. and Sudderth, W. D. (1974). Continuous-time gambling problems. Adv. Appl. Prob. 6, 651–665.Google Scholar
[7] Ikeda, N. and Watanabe, S. (1981) Stochastic Differential Equations and Diffusion Processes. Kodansha, Tokyo.Google Scholar
[8] Klawe, M. M. (1979) Optimal strategies for a fair betting game. Discrete Appl. Math. 1, 105–115.Google Scholar
[9] Knight, F. B. (1971) A reduction of continuous square-integrable martingales to Brownian motion. In Martingales, ed. Dinges, H., Lecture Notes in Mathematics 190, Springer-Verlag, Berlin.Google Scholar
[10] Knight, F. B. (1981) Essentials of Brownian Motion and Diffusion. Mathematical Surveys. Number 18. American Mathematical Society, Providence, Rhode Island.Google Scholar
[11] Pestien, V. C. and Sudderth, W. D. (1985) Continuous-time red and black: how to control a diffusion to a goal. Math. Operat. Res. 10, 599–611.Google Scholar
[12] Spencer, J. (1986) Balancing vectors in the max norm. Combinatorica 6, 55–65.Google Scholar
[13] Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag. New York.Google Scholar