IMT Institutional Repository: No conditions. Results ordered -Date Deposited. 2024-03-29T14:54:18ZEPrintshttp://eprints.imtlucca.it/images/logowhite.pnghttp://eprints.imtlucca.it/2013-11-06T11:50:13Z2016-04-06T09:55:08Zhttp://eprints.imtlucca.it/id/eprint/1876This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/18762013-11-06T11:50:13ZTime evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions 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. Boris PodobnikKaushik MatiaAlessandro Chessaalessandro.chessa@imtlucca.itPlamen Ch. IvanovYoungki LeeH. Eugene Stanley2013-11-06T11:44:31Z2013-11-20T09:00:49Zhttp://eprints.imtlucca.it/id/eprint/1875This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/18752013-11-06T11:44:31ZSystems with correlations in the variance: Generating power law tails in probability distributionsWe study how the presence of correlations in physical variables contributes to the form of probability distributions. We investigate a process with correlations in the variance generated by i) a Gaussian or ii) a truncated Lévy distribution. For both i) and ii), we find that due to the correlations in the variance, the process "dynamically" generates power law tails in the distributions, whose exponents can be controlled through the way the correlations in the variance are introduced. For ii), we find that the process can extend a truncated distribution beyond the truncation cutoff, which leads to a crossover between a Lévy stable power law and the present "dynamically generated" power law. We show that the process can explain the crossover behavior recently observed in the S&P500 stock index.Boris PodobnikPlamen Ch. IvanovYoungki LeeAlessandro Chessaalessandro.chessa@imtlucca.itH. Eugene Stanley