Asymptotic optimality and asymptotic equipartiton properties of log- optimum investment. (English) Zbl 0642.90016

Algoet, Paul H.; Cover, Thomas M.

Ann. Probab. 16, No. 2, 876-898 (1988).

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. 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 \(k\to \infty\).

MSC:

91B28	Finance etc. (MSC2000)
91B62	Economic growth models

Keywords:

maximum capital growth rate; log-optimum investment policies; asymptotic growth rate

Citations:

Zbl 0109.368