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An extended Fatou equation and continuous-time gambling

Published online by Cambridge University Press:  01 July 2016

Victor C. Pestien
Victor C. Pestien*
Affiliation:
The University of Miami
*
∗Postal address: Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, FL 33124, U.S.A.
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Abstract

Let be an optional stochastic process and let be the directed set of almost-surely-finite stopping times. If lim , then

This equality is employed in studying the continuous-time gambler's problem, as formulated by Heath and Sudderth (1974). The optimal return function is shown to be upper semi-analytic and excessive whenever the utility function is Borel.

Keywords

GAMBLING THEORYSTOCHASTIC CONTROL
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 2 , June 1982 , pp. 309 - 323
Copyright
Copyright © Applied Probability Trust 1982 

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References

Bertsekas, D. P. and Shreve, S. E. (1978) Stochastic Optimal Control: the Discrete Time Case. Academic Press, New York.Google Scholar
Chen, R. (1976) Some inequalities for randomly stopped variables with applications to pointwise convergence. Z. Wahrscheinlichkeitsth. 36, 75–83.Google Scholar
Dellacherie, C. and Meyer, P. A. (1975) Probabilités et potentiel. Hermann, Paris.Google Scholar
Dubins, L. E. and Freedman, D. (1966) On the expected value of a stopped martingale. Ann. Math. Statist. 37, 1505–1509.Google Scholar
Dubins, L. E. and Savage, L. J. (1965), (1976) Inequalities for Stochastic Processes: How to Gamble if You Must. Dover, New York.Google Scholar
Dubins, L. E. and Sudderth, W. D. (1977a) Countably additive gambling and optimal stopping. Z. Wahrscheinlichkeitsth. 41, 59–72.Google Scholar
Dubins, L. E. and Sudderth, W. D. (1977b) Persistently e-optimal strategies. Math. Operat. Res. 2, 125–134.Google Scholar
Engelbert, A. and Engelbert, H. J. (1979) Optimal stopping and almost sure convergence of random sequences. Z. Wahrscheinlichkeitsth. 48, 309–325.Google Scholar
Heath, D. C. and Sudderth, W. D. (1974) Continuous-time gambling problems. Adv. Appl. Prob. 6, 651–665.Google Scholar
Mertens, J.-F. (1972) Théorie des processus stochastiques généraux, applications aux surmartingales. Z. Wahrscheinlichkeitsth. 22, 45–68.Google Scholar
Pestien, V. C. (1980) Stopping-time-indexed Convergence and Measurable Gambling. , Dept. of Math., University of California, Berkeley.Google Scholar
Sudderth, W. D. (1970) A ‘Fatou Equation’ for randomly stopped variables. Technical Report No. 137, University of Minnesota.Google Scholar
Sudderth, W. D. (1971) A ‘Fatou equation’ for randomly stopped variables. Ann. Math. Statist. 42, 2143–2146.Google Scholar
Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar