Sequential Portfolio Selection under Latent Side Information-Dependence Structure: Optimality and Universal Learning Algorithms

This paper investigates the investment problem of constructing an optimal no-short sequential portfolio strategy in a market with a latent dependence structure between asset prices and partly unobservable side information, which is often high-dimensional. The results demonstrate that a dynamic strategy, which forms a portfolio based on perfect knowledge of the dependence structure and full market information over time, may not grow at a higher rate infinitely often than a constant strategy, which remains invariant over time. Specifically, if the market is stationary, implying that the dependence structure is statistically stable, the growth rate of an optimal dynamic strategy, utilizing the maximum capacity of the entire market information, almost surely decays over time into an equilibrium state, asymptotically converging to the growth rate of a constant strategy. Technically, this work reassesses the common belief that a constant strategy only attains the optimal limiting growth rate of dynamic strategies when the market process is identically and independently distributed. By analyzing the dynamic log-optimal portfolio strategy as the optimal benchmark in a stationary market with side information, we show that a random optimal constant strategy almost surely exists, even when a limiting growth rate for the dynamic strategy does not. Consequently, two approaches to learning algorithms for portfolio construction are discussed, demonstrating the safety of removing side information from the learning process while still guaranteeing an asymptotic growth rate comparable to that of the optimal dynamic strategy.