Itô’s Excursion Theory, the Hurst Coefficient, and Fractional Excursions in Finance

Abstract

In this paper, we investigate how Ito’s excursion theory can be usefully applied to economic time series data (Ito 2007). We relate excursion theory to geometric and fractional Brownian motion and the Hurst coefficient. We then calculate the Hurst coefficient for all stocks on the DOW 30, S&P 500 and Russell 2000, showing the distribution of Hurst measures and relating them statistically to excursions. In doing so we provide a nice and intuitive link between Brownian motion and excursions, an application and consequence that we have not seen before.