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

TL;DR

This paper introduces a financial model incorporating memory effects via fractional calculus and fractional Brownian motion, analyzing phase behavior and dispersion relations to understand complex market dynamics.

Contribution

It presents a novel application of fractional Langevin equations with colored noise to model financial systems, exploring phase behavior and potential glass phases.

Findings

Potential identification of anomalous marginal glass phase

Analysis of phase behavior and dispersion relations in financial models

Use of physics-based methods for financial system analysis

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.