Anomalous scaling of stochastic processes and the Moses effect

Area of Science:

• Stochastic processes
• Time series analysis
• Financial mathematics

Background:

• Stochastic processes typically exhibit normal distributions with scaling exponents of 1/2.
• Anomalous scaling occurs when distributions are non-normal and scaling exponents differ from 1/2.
• Understanding anomalous scaling origins is crucial for accurate process quantification.

Purpose of the Study:

• To identify and quantify the origins of anomalous scaling in stochastic processes.
• To define and relate scaling exponents (Noah, Joseph, Moses effects) to the Hurst exponent.
• To analyze financial time series data for anomalous scaling mechanisms.

Main Methods:

• Definition of scaling exponents for stationary (Noah, Joseph) and nonstationary (Moses) increments.
• Development of time series analysis methods for independent exponent measurement.
• Application of methods to simple stochastic processes and intraday financial data.

Main Results:

• Autocorrelations (Joseph effect) and infinite variance (Noah effect) cause anomalous scaling in stationary processes.
• Nonstationary increments (Moses effect) also lead to anomalous scaling.
• Intraday financial time series data display anomalous scaling exclusively due to the Moses effect.

Conclusions:

• The Moses effect is the sole contributor to anomalous scaling in financial time series.
• The Joseph exponent, not the Hurst exponent, is the appropriate measure for testing the efficient market hypothesis.
• This research provides a framework for analyzing and understanding anomalous scaling in various stochastic systems.