Stochastic processes with power-law stability and a crossover in power-law correlations

Motivated by the goal of finding a more accurate description of the empirically observed dynamics of financial fluctuations, we propose a stochastic process that yields three statistical properties: (i) short-range autocorrelations in the index changes, (ii) long-range correlations in the absolute values of the index changes, with a crossover between two power-law regimes at approximately one week, and (iii) power-law stability in the tails of the probability distributions of the index changes. We find that this stochastic process can surprisingly well reproduce statistical properties observed in the high-frequency data of the S&P 500 stock index.