Asymptotic arbitrage in fractional mixed markets
Volume 5, Issue 4 (2018), pp. 415–428
Pub. online: 20 August 2018
Type: Research Article
Open Access
Open Access
Received
18 May 2018
18 May 2018
Revised
12 July 2018
12 July 2018
Accepted
12 July 2018
12 July 2018
Published
20 August 2018
20 August 2018
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.
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