Anomalous scaling of stochastic processes and the Moses effect*

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. 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 illustrate each effect. Analysis of Intraday Financial time series data reveals that its anomalous scaling is due only to the Moses effect and the lack of Joseph effect implies that the market is efficient.