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

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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