St. Petersburg Portfolio Games

Abstract

We investigate the performance of the constantly rebalanced portfolios, when the random vectors of the market process { X _i } are independent, and each of them distributed as (X ⁽¹⁾, X ⁽²⁾, ..., X ^(d), 1), d ≥ 1, where X ⁽¹⁾, X ⁽²⁾, ..., X ^(d) are nonnegative iid random variables. Under general conditions we show that the optimal strategy is the uniform: (1/d, ..., 1/d, 0), at least for d large enough. In case of St. Petersburg components we compute the average growth rate and the optimal strategy for d = 1,2. In order to make the problem non-trivial, a commission factor is introduced and tuned to result in zero growth rate on any individual St. Petersburg components. One of the interesting observations made is that a combination of two components of zero growth can result in a strictly positive growth. For d ≥ 3 we prove that the uniform strategy is the best, and we obtain tight asymptotic results for the growth rate.