Distinguishing between fractional Brownian motion with random and constant Hurst exponent using sample...

Area of Science:

• Complex Systems Dynamics
• Statistical Physics
• Time Series Analysis

Background:

• Fractional Brownian motion (FBM) models complex system dynamics using the Hurst exponent.
• Classical FBM may not capture experimental data where the anomalous diffusion exponent varies per trajectory.
• FBM with a random Hurst exponent is a proposed extension to address these limitations.

Purpose of the Study:

• To develop a method for distinguishing between FBM with a constant Hurst exponent and FBM with a random Hurst exponent.
• To analyze the probabilistic properties of quadratic form statistics for hypothesis testing.
• To apply the developed methodology to real-world financial and biological datasets.

Main Methods:

• Examined the sample autocovariance function and empirical anomaly measure statistics.
• Utilized correlation properties inherent to the FBM models.
• Developed a testing procedure based on these statistics to differentiate between constant and random Hurst exponent models.
• Considered two-point and beta distributions for the random Hurst exponent.

Main Results:

• Presented analytical and simulation results demonstrating the effectiveness of the proposed testing procedure.
• Showcased the ability to differentiate between the two FBM models based on statistical properties.
• Validated the methodology on financial market data and single particle tracking experiments.

Conclusions:

• The developed statistical testing procedure effectively distinguishes between FBM with constant and random Hurst exponents.
• The methodology provides a valuable tool for analyzing complex systems exhibiting trajectory-dependent anomalous diffusion.
• The approach has practical applications in fields like finance and biophysics.