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Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions

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)

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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.

Item Type: Article
Identification Number: 10.1016/S0378-4371(01)00390-9
Uncontrolled Keywords: Random walks; Stochastic processes; Fluctuation phenomena; Central limit theory
Subjects: Q Science > QC Physics
Research Area: Economics and Institutional Change
Depositing User: Ms T. Iannizzi
Date Deposited: 06 Nov 2013 11:50
Last Modified: 06 Apr 2016 09:55
URI: http://eprints.imtlucca.it/id/eprint/1876

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