Time 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 ARGARCH process, where AR denotes the autoregressive process which we use to account for the short-range correlations in the index changes and GARCH den otes the generalized autoregressive conditional heteroskedastic process whi ch takes into account the long-range correlations in the variance. We study the AR+GARCH process with an initial distribution of truncated Levy form. We find that this process generates a new probability distribution with a c rossover from a Levy stable power law to a power law with an exponent outsi de the Levy range, beyond the truncation cutoff, We analyze the sum of n va riables 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 obs erved in the probability distribution of the S&P500 index. We find that thi s power-law stability depends on the characteristic scale in the correlatio ns. 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 it as observed in the data. (C) 2001 Elsevier Science B.V. All r ights reserved.