The Nature of Heavy Tails in Statistics of Random Processes and Predictability of Market Prices

摘要

The talk consists of two parts. In the first part the nature of the aged problem of heavy tails [power-law decay of tails of probability densities (PD) for sums of random independent variables] is revealed. The exact expression for the probability density p_N (x) for sums of a finite number N of random independent terms is obtained. It is shown that the very tail of p_N (x) has a Gaussian form if and only if all the random terms are distributed according to the Gauss Law. In all other cases the tail for p_N (x) differs from the Gaussian. If the variances of random terms diverge the non-Gaussian tail is related to a Levy distribution for p_N (x). However, the tail is not Gaussian even if the variances are finite. In the latter case p_N (x) has two different asymptotics. At small and moderate values of x the distribution is Gaussian. At large x the non-Gaussian tail arises. The crossover between the two asymptotics occurs at x proportional to N. Thus, the non-Gaussian tail exists at finite N only. In the limit N tends to infinity the origin of the tail is shifted to infinity, i. e., the tail vanishes. Depending on the particular type of the distribution of the random terms the non-Gaussian tail may decay either slower than the Gaussian, or faster than it. A number of particular examples is discussed in detail. In the second part the dynamics of actual market prices is analyzed based upon the theory of dynamical chaos. It is shown that the dynamics is typical for that of chaotic dynamical systems. A number of particular examples (exchange rates USD vs. JPY, XAU vs. USD, oil, etc.) is considered in detail. The topological structure of the corresponding phase space for the price dynamics is reconstructed. It allows predicting the price dynamics in future with high accuracy. The main advantage of the approach is that, as long as the forecast period lies below the so-called prediction horizon, the prediction error does not increase in the course of time.