# Anomalous Scaling of Stochastic Processes and the Moses Effect

## TL;DR

This paper investigates the causes of anomalous scaling in stochastic processes, introducing the Moses effect as a new mechanism, and applies the analysis to financial data to identify its role in market fluctuations.

## Contribution

The paper defines and relates the Moses effect to other known effects, providing methods to measure each and demonstrating their relevance in financial time series analysis.

## Key findings

- Financial data's anomalous scaling is due to the Moses effect.
- The Joseph exponent, not the Hurst exponent, tests market efficiency.
- Methods enable independent measurement of scaling effects.

## 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 $\frac{1}{2}$ 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 $\it{Joseph}$ $\it{effect}$ the $\it{Noah}$ $\it{effect}$, 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 $\it{Moses}$ $\it{effect}$. 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.