relation: http://eprints.imtlucca.it/1876/
title: Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions 
creator: Podobnik, Boris
creator: Matia, Kaushik
creator: Chessa, Alessandro
creator: Ivanov, Plamen Ch.
creator: Lee, Youngki
creator: Stanley, H. Eugene
subject: QC Physics
description: We model the time series of the S&P500 index by a combined process, the AR+GARCH process, where {AR} denotes the autoregressive process which we use to account for the short-range correlations in the index changes and {GARCH} denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long-range correlations in the variance. We study the AR+GARCH process with an initial distribution of truncated Lévy form. We find that this process generates a new probability distribution with a crossover from a Lévy stable power law to a power law with an exponent outside the Lévy range, beyond the truncation cutoff. We analyze the sum of n variables of the AR+GARCH process, and find that due to the correlations the AR+GARCH process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n—a feature which is observed in the probability distribution of the S&P500 index. We find that this power-law stability depends on the characteristic scale in the correlations. We also find that inclusion of short-range correlations through the {AR} process is needed to obtain convergence to a limiting Gaussian distribution for large n as observed in the data. 
publisher: Elsevier
date: 2001
type: Article
type: PeerReviewed
identifier:   Podobnik, Boris and Matia, Kaushik and Chessa, Alessandro and Ivanov, Plamen Ch. and Lee, Youngki and Stanley, H. Eugene  Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions.  Physica A: Statistical Mechanics and its Applications, 300 (1–2).  300 - 309.  ISSN 0378-4371      (2001)  
relation: http://www.sciencedirect.com/science/article/pii/S0378437101003909
relation: 10.1016/S0378-4371(01)00390-9