Multifractality in time series

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Abstract
We apply the concepts of multifractal physics to financial time series in order to characterize the onset of crash for the Standard & Poor 500 (S&P500) stock index x(t). It is found that within the framework of multifractality, the `analogous' specific heat of the S&P500 discrete price index displays a shoulder to the right of the main peak for low time-lag values. For decreasing T, the presence of the shoulder is a consequence of the peaked, temporal x(t + T)-x(t) fluctuations in this regime. For large time lags (T > 80), we have found that Cq displays typical features of a classical phase transition at a critical point. An example of such dynamic phase transition in a simple economic model system, based on a mapping with multifractality phenomena in random multiplicative processes, is also presented by applying former results obtained with a continuous probability theory for describing scaling measures.
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