Detecting subtle deviations in Brownian motion representations driven by a Schauder basis

Area of Science:

• Stochastic Processes
• Mathematical Finance
• Time Series Analysis

Background:

• Brownian motion is fundamental in modeling random phenomena.
• Distinguishing true Brownian motion from surrogates is crucial for accurate analysis.
• Schauder representation offers a method for constructing stochastic processes.

Purpose of the Study:

• To construct and analyze surrogate stochastic processes.
• To investigate the fidelity of non-Gaussian Schauder-based constructions compared to Brownian motion.
• To identify subtle deviations from true Brownian motion in generated sample paths.

Main Methods:

• Utilizing the Schauder representation for stochastic process construction.
• Employing non-Gaussian distributions (beta, uniform) for Schauder coefficients.
• Applying entropy-based measures and spectral variance analysis for deeper evaluation.

Main Results:

• Non-Gaussian surrogates preserved properties like quadratic variation and covariance structure.
• Only Gaussian-based constructions maintained consistent spectral behavior and randomness.
• Subtle, yet significant, deviations from Brownian motion were observed in non-Gaussian variants.

Conclusions:

• The Schauder representation can generate realistic Brownian motion surrogates.
• Gaussian-based constructions are superior in replicating true Brownian motion characteristics.
• Advanced analytical techniques are necessary to detect subtle differences in stochastic processes.