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Portfolio choice and the Bayesian Kelly criterion

Part of: Decision theory Markov processes Stochastic processes

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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