Modeling a Financial System with Memory via Fractional Calculus and Fractional Brownian Motion

ArXiv ID: 2406.19408 “View on arXiv”

Authors: Unknown

Abstract

Financial markets have long since been modeled using stochastic methods such as Brownian motion, and more recently, rough volatility models have been built using fractional Brownian motion. This fractional aspect brings memory into the system. In this project, we describe and analyze a financial model based on the fractional Langevin equation with colored noise generated by fractional Brownian motion. Physics-based methods of analysis are used to examine the phase behavior and dispersion relations of the system upon varying input parameters. A type of anomalous marginal glass phase is potentially seen in some regions, which motivates further exploration of this model and expanded use of phase behavior and dispersion relation methods to analyze financial models.

Keywords: Fractional Langevin Equation, Fractional Brownian Motion, Stochastic Volatility, Colored Noise, Phase Behavior, Equities

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is saturated with advanced mathematics, including extensive fractional calculus, stochastic processes, and physics-based phase analysis, indicating high complexity. However, the methodology is entirely theoretical with no backtesting, implementation, or data-driven validation, placing it firmly in the theoretical ‘Lab Rats’ category.

  flowchart TD
    A["Research Goal<br>Model Financial Volatility<br>with Memory via FBM"] --> B
    subgraph B ["Methodology"]
        B1["Fractional Langevin Eq.<br>with Colored Noise"]
        B2["Physics-Based Analysis<br>Phase & Dispersion"]
    end
    B --> C["Data/Inputs<br>Stochastic Volatility Parameters"]
    C --> D["Computational Process<br>Simulation & Analytical Solution"]
    D --> E["Outcomes<br>Anomalous Marginal Glass Phase"]
    E --> F["Conclusion<br>Validates FBM for<br>Memory-Dependent Finance"]