Vygintas Gontis

We aim to explain the phenomenon of long‐range memory in social systems by nonlinear interactions of agents resulting in the macroscopic description by nonlinear stochastic differential equations. Thus we deal with models of social systems, empirical analysis of data and theoretical consideration of the problem. The statistics of return and trading activity in the financial markets is the first task of our research. The scope of this research is of the highest importance as slowly decaying auto‐correlation and 1/f noise occurs in Markov processes defined as memoryless. This means that in many real systems observed property of long‐range memory may be spurious and might origin not from the correlations in stochastic noise. Thus the modeling of many real systems by fractional Brownian motion might be inappropriate. The specific statistical tests and criterions are needed to identify alternative methods for the modeling of long‐range memory in real social systems. The identification of such criterions is the most significant task of our research.

It is well known that for the fBm first passage time PDF is of the power‐law form with exponent dependent on Hurst exponent, as 2‐H, while for the one‐dimensional Markov processes power‐law exponent is always 3/2. Consequently a detailed empirical analysis of the first passage times in return volatility and trading activity time series could be used to identify the true nature of the observed long‐range memory effect. In this contribution we will present results of empirical study for high frequency time series in FOREX exhibiting exponent of first passage time PDF close to 3/2, with all deviations explained by the extraneous noises: order flow dynamics, daily seasonality and high‐frequency speculative trading.