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)

## Abstract

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 |
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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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