Hostname: page-component-6766d58669-nqrmd Total loading time: 0 Render date: 2026-05-24T12:53:56.531Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous-time gambling problems

Published online by Cambridge University Press:  01 July 2016

David C. Heath  and
William D. Sudderth
David C. Heath
Affiliation:
University of Minnesota
William D. Sudderth
Affiliation:
University of Minnesota
Get access Rights & Permissions [Opens in a new window]

Abstract

An abstract gambler's problem is formulated in a continuous-time setting and analogues are proved for some of the discrete-time results of Dubins and Savage in their book How to Gamble if You Must. Applications are made to problems of controlling a Brownian motion process.

Keywords

GAMBLING THEORYSTOCHASTIC CONTROLDYNAMIC PROGRAMMINGOPTIMAL STRATEGIESCONTINUOUS-TIME PROCESSESBROWNIAN MOTION

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 4 , December 1974 , pp. 651 - 665
Copyright
Copyright © Applied Probability Trust 1974 

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.)

Article purchase

Temporarily unavailable

References

[1] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
[2] Dubins, L. E. and Savage, L. J. (1965) How to Gamble If You Must. McGraw – Hill, New York.Google Scholar
[3] Dynkin, E. B. (1965) Markov Processes. Springer – Verlag, Berlin.Google Scholar
[4] Fakeev, A. G. (1970) Optimal stopping rules for stochastic processes with continuous parameter. Theor. Probability Appl. 15, 324–331.CrossRefGoogle Scholar
[5] Kolmogorov, A. N. (1956) On Skorohod convergence. Theor. Probability Appl. 1, 215–222.Google Scholar
[6] Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D [0, ∞). J. Appl. Prob. 10, 109–121.CrossRefGoogle Scholar
[7] Mackey, G. W. (1957) Borel structures in groups and their duals. Trans. Amer. Math. Soc. 85, 134–165 CrossRefGoogle Scholar
[8] Meyer, P. A. (1966) Probability and Potentials. Blaisdell Publishing, Waltham, Mass.Google Scholar
[9] Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.Google Scholar
[10] Rishel, R. (1970) Necessary and sufficient conditions for continuous time stochastic optimal control. SIAM J. Control 8, 559–571.Google Scholar
[11] Strauch, R. E. (1967) Measurable gambling houses. Trans. Amer. Math. Soc. 126, 64–72.Google Scholar
[12] Streibel, C. (1974) Martingale conditions for the optimal control of continuous time stochastic systems. University of Minnesota preprint.Google Scholar
[13] Sudderth, W. D. (1971) A ‘Fatou equation’ for randomly stopped variables. Ann. Math. Statist. 42, 2143–2146.Google Scholar