Fractional Langevin model of memory in financial markets

The separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the Langevin equation. When the microscopic time scale diverges, the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion; in particular, we obtain a fractional-differential stochastic equation of motion. After decades of statistical analysis of financial time series certain "stylized facts" have emerged, including the statistics of stock price fluctuations having "fat tails" and their linear correlations in time being exceedingly short lived. On the other hand, the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law. One explanation of this long-term memory is that it is a consequence of the time-scale separation between "microscopic" and "macroscopic" economic variables. We propose a fractional Langevin equation as a dynamical model of the observed memory in financial time series.