Fractional Ornstein-Uhlenbeck processes driven by stable Lévy motion in finance

Abstract

In this study we consider the fractional Ornstein-Uhlenbeck processes driven by α-stable Levy motion as a model of financial time series. This type models have ability flexible modeling for real data. The α-stable distributions have infinite second moment, for this reason, we can use co-difference to describe the dependence structure of series. Fractional Lévy process is a natural generalization of the integral representation of fractional Brownian motion. Fractional Lévy driven Ornstein-Uhlenbeck process is the unique stationary solution of the corresponding Langevin equation. Auto-covariance function of this stationary solution is like that of a power function. Futhermore increment process exhibit long-range dependence. The effectiveness of the analytical predictions is checked via analysis of the parameters and is tested on a data set of financial indices. In this study We apply the our model to Exon Mobil Corp (XOM) stock returns and Standard and Poor 500(S&P500) index return data. We illustrate that there is a good agreement between the empirical data and the theoretical description. © EuroJournals Publishing, Inc. 2010.