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Portfolio choice and the Bayesian Kelly criterion

Part of: Stochastic processes Markov processes Decision theory

Published online by Cambridge University Press:  01 July 2016

Sid Browne  and
Ward Whitt
Sid Browne*
Affiliation:
Columbia University
Ward Whitt*
Affiliation:
AT&T Bell Laboratories
*
∗ Postal address: 402 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027, USA.
∗∗ Postal address: AT&T Bell Laboratories, Room 2C-178, Murray Hill NJ 07974-0636, USA.
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Abstract

We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: the optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the diffusion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. We then find conditions under which the discrete-time process converges to a diffusion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limit diffusion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuous-time gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the financial value of randomness, the financial gain of perfect information and the financial cost of learning in the Bayesian problem.

Keywords

BETTING SYSTEMSPROPORTIONAL GAMBLINGKELLY CRITERIONPORTFOLIO THEORYLOGARITHMIC UTILITYRANDOM WALKS IN A RANDOM ENVIRONMENTKIEFER PROCESSTIME-CHANGED BROWNIAN MOTIONCONJUGATE PRIORSBAYESIAN CONTROL

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62C10: Bayesian problems; characterization of Bayes procedures 60J60: Diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 1145 - 1176
Copyright
Copyright © Applied Probability Trust 1996 

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References

Algoet, P. H. and Cover, T. M. (1988) Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Ann. Prob. 16, 876–898.CrossRefGoogle Scholar
Barron, A. and Cover, T. M. (1988) A bound on the financial value of information. IEEE Trans. Info. Theory 34, 1097–1100.Google Scholar
Bell, R. M. and Cover, T. M. (1980) Competitive optimality of logarithmic investment. Math. Operat. Res. 5, 161–166.CrossRefGoogle Scholar
Bellman, R. and Kalaba, R. (1957) On the role of dynamic programming in statistical communication theory. IRE Trans. Info. Theory IT3, 197–203.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Breiman, L. (1961) Optimal gambling systems for favorable games. Fourth Berkeley Symp. Math. Stat. Prob. 1, 65–78.Google Scholar
Cover, T. M. (1991) Universal portfolios. Math. Finance 1, 1–29.Google Scholar
Cover, T. M. and Gluss, D. H. (1986) Empirical Bayes stock market portfolios. Adv. Appl. Math. 7, 170–181.Google Scholar
Cover, T. M. and Thomas, J. (1991) Elements of Information Theory. Wiley, New York.Google Scholar
Csörgö, M. and Révész, P. (1981) Strong Approximations in Probability and Statistics. Academic Press, New York.Google Scholar
Degroot, M. H. (1970) Optimal Statistical Decisions. McGraw-Hill, New York.Google Scholar
Ethier, S. N. (1988) The proportional bettor's fortune. Proc. 5th Int. Conf. on Gambling and Risk Taking 4, 375–383.Google Scholar
Ethier, S. N. and Tavare, S. (1983) The proportional bettor's return on investment. J. Appl. Prob. 20, 563–573.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2. 2nd edn. Wiley, New York.Google Scholar
Finkelstein, M. and Whitely, R. (1981) Optimal strategies for repeated games. Adv. Appl. Prob. 13, 415–428.Google Scholar
Fleming, W. H. and Rishel, R. W. (1975) Deterministic and Stochastic Control. Springer, Berlin.Google Scholar
Gottlieb, G. (1985) An optimal betting strategy for repeated games. J. Appl. Prob. 22, 787–795.Google Scholar
Hakansson, N. (1970) Optimal investment and consumption strategies under risk for a class of utility functions. Econometrica 38, 587–607.Google Scholar
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
Jamishidian, F. (1992) Asymptotically optimal portfolios. Math. Finance 2, 131–150.Google Scholar
Kallianpur, G. (1980) Stochastic Filtering Theory. Springer, Berlin.Google Scholar
Karatzas, I. (1989) Optimization problems in the theory of continuous trading. SIAM J. Control Optim. 7, 1221–1259.Google Scholar
Karatzas, I. and Shreve, S. (1988) Brownian Motion and Stochastic Calculus. Springer, Berlin.Google Scholar
Kelly, J. (1956) A new interpretation of information rate. Bell Sys. Tech. J. 35, 917–926.CrossRefGoogle Scholar
Kurtz, T. G. and Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Prob. 19, 1035–1070.CrossRefGoogle Scholar
Kushner, H. J. and Dupuis, P. G. (1992) Numerical Methods for Stochastic Control Problems in Continuous-Time. Springer, Berlin.Google Scholar
Latane, H. (1959) Criteria for choice among risky assets. J. Polit. Econ. 35, 144–155.CrossRefGoogle Scholar
Merton, R. (1971) Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373–413.Google Scholar
Merton, R. (1990) Continuous Time Finance. Blackwell, Oxford.Google Scholar
Pestien, V. C. and Sudderth, W. D. (1985) Continuous-time red and black: how to control a dffusion to a goal. Math. Operat. Res. 10, 599–611.Google Scholar
Révész, P. (1990) Random Walk in Random and Non-Random Environments. World Scientific, Singapore.Google Scholar
Thorp, E. O. (1969) Optimal gambling systems for favorable games. Rev. Int. Statist. Inst. 37, 273–293.CrossRefGoogle Scholar
Thorp, E. O. (1971) Portfolio choice and the Kelly criterion. Bus. Econ. Statist Sec., Proc. Amer. Stat. Assoc. 215–224.Google Scholar