Exponentially concave functions and high dimensional stochastic portfolio theory

摘要

We construct an explicit example of asymptotic short term relative arbitrage. Specifically, for every n we assume an n dimensional semimartingale market model that starts from a heavy-tailed initial position in the unit simplex and impose weak assumptions on its volatility. We then construct a sequence of portfolios, one for each dimension, that outperform the market portfolio in dimension n by an amount M-n by time delta(n) with a probability at least 1 - q(n). Here M-n -> infinity exponentially fast in n and delta(n), q(n) decrease to zero. Moreover, these portfolios never underperform below a pre-specified lower bound. The key fact is that it is possible to construct a sequence of exponentially concave functions on the unit simplex of increasing concavity because the typical diameter of the unit simplex in dimension n is O (1/root n). (C) 2018 Elsevier B.V. All rights reserved.