%0 Journal Article
%@ 0378-4371
%A Podobnik, Boris
%A Matia, Kaushik
%A Chessa, Alessandro
%A Ivanov, Plamen Ch.
%A Lee, Youngki
%A Stanley, H. Eugene
%D 2001
%F eprints:1876
%I Elsevier
%J Physica A: Statistical Mechanics and its Applications
%K Random walks; Stochastic processes; Fluctuation phenomena; Central limit theory
%N 1–2
%P 300 - 309
%T Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions 
%U http://eprints.imtlucca.it/1876/
%V 300
%X 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.