Asymptotic arbitrage in fractional mixed markets

TL;DR

This paper investigates the presence of strong asymptotic arbitrage opportunities in mixed fractional markets, where the market model combines fractional Brownian motion and standard Brownian motion, using large financial market techniques.

Contribution

It demonstrates the existence of strong asymptotic arbitrage in mixed fractional markets as the scaling factor approaches zero, extending the understanding of arbitrage in complex stochastic models.

Findings

Strong asymptotic arbitrage exists when the scaling factor tends to zero.

The analysis uses Gaussian measure separation and relative entropy.

The results apply to markets modeled by mixed fractional Brownian motion.

Abstract

We consider a family of mixed processes given as the sum of a fractional Brownian motion with Hurst parameter $H\in(3/4,1)$ and a multiple of an independent standard Brownian motion, the family being indexed by the scaling factor in front of the Brownian motion. We analyze the underlying markets with methods from large financial markets. More precisely, we show the existence of a strong asymptotic arbitrage (defined as in Kabanov and Kramkov [Finance Stoch. 2(2), 143--172 (1998)]) when the scaling factor converges to zero. We apply a result of Kabanov and Kramkov [Finance Stoch. 2(2), 143--172 (1998)] that characterizes the notion of strong asymptotic arbitrage in terms of the entire asymptotic separation of two sequences of probability measures. The main part of the paper consists of proving the entire separation and is based on a dichotomy result for sequences of Gaussian measures and the concept of relative entropy.