Anomalous scaling of stochastic processes and the Moses effect

Abstract

The state of a stochastic process evolving over a time t is typically assumed to lie on a normal distribution whose width scales like t 1 2 . However, processes where the probability distribution is not normal and the scaling exponent differs from 1 2 are known. The search for possible origins of such "anomalous" scaling and approaches to quantify them are the motivations for the work reported here. In processes with stationary increments, where the stochastic process is time-independent, auto-correlations between increments and infinite variance of increments can cause anomalous scaling. These sources have been referred to as the Joseph effect the Noah effect, respectively. If the increments are non-stationary, then scaling of increments with t can also lead to anomalous scaling, a mechanism we refer to as the Moses effect. Scaling exponents quantifying the three effects are defined and related to the Hurst exponent that characterizes the overall scaling of the stochastic process. Methods of time series analysis that enable accurate independent measurement of each exponent are presented. Simple stochastic processes are used to illustrate each effect. Intraday financial time series data is analyzed, revealing that its anomalous scaling is due only to the Moses effect. In the context of financial market data, we reiterate that the Joseph exponent, not the Hurst exponent, is the appropriate measure to test the efficient market hypothesis.

Inferring characteristic timescales from the effect of autoregressive dynamics on detrended fluctuation analysis

“…This should not be confused with other anomalous statistical properties. Anomalous diffusion can have three root causes [23,24]. The correlation time can be infinite, the second moment of the increments can diverge or the system can be non-stationary.…”

Anomalous diffusion and the Moses effect in an aging deterministic model

“…Many experimental systems, however, are known to diffuse anomalously. Examples include cold atoms in dissipative optical lattices [1,2], motion in a crowded environment such as the cytoplasm of biological cells [3][4][5], blinking quantum dots [6][7][8], and intra-day trades in financial markets [9][10][11]. Understanding the nature of the dynamics of these systems that leads to anomalous diffusion is a topic of intense interest.…”

Objective comparison of methods to decode anomalous diffusion

“…Here, difficulties arise because the calculation of the MSD is not very informative, since different models provide curves with the same scaling exponent. Other statistical parameters have been proposed for this task and algorithms based on the combination of several estimators allow to distinguish between pairs of models [21][22][23][24] , but there is no general consensus on how to unambiguously determine the underlying diffusion model from a trajectory.…”