Asymptotic optimality and asymptotic equipartiton properties of log- optimum investment
type: Item
instance of: scholarly article
title: Asymptotic optimality and asymptotic equipartiton properties of log- optimum investment
zbMATH Open document ID: 0642.90016
author: Paul H. Algoet
author: Thomas M. Cover
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: We ask how an investor (with knowledge of the past) should distribute his funds over various investment opportunities to maximize the growth rate of his compounded capital. \textit{L. Breiman} [Proc. 4th Berkeley Symp. Math. Stat. Probab. 1, 65-78 (1961; Zbl 0109.368)] answered this question when the stock returns for successive periods are independent, identically distributed random vectors. We prove that maximizing conditionally expected log return given currently available information at each stage is asymptotically optimum, with no restrictions on the distribution of the market process. If the market is stationary ergodic, then the maximum capital growth rate is shown to be a constant almost surely equal to the maximum expected log return given the infinite past. Indeed, log-optimum investment policies that at time n look at the n-past are sandwiched in asymptotic growth rate between policies that look at only the k-past and those that look at the infinite past, and the sandwich closes as \(k\to \infty\).
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 91B62
zbMATH DE Number: 4047384
zbMATH Keywords: maximum capital growth rate
zbMATH Keywords: log-optimum investment policies
zbMATH Keywords: asymptotic growth rate
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991793
OpenAlex ID: W2040759104
DOI: 10.1214/AOP/1176991793
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description: scientific article; zbMATH DE number 4047384