Non-Linear and Meta-Stable Dynamics in Financial Markets: Evidence from High Frequency Crypto Currency Market Makers

2025-09-02

AI Summary

This study challenges the conventional linear and quadratic potential diffusion models by investigating whether higher-order nonlinear (cubic or higher) drift terms dominate financial market dynamics. Using high-frequency cryptocurrency market-maker data, we develop a multiscale diffusion process model incorporating nonlinear drift, capital flow modeling, and nonlinear time-series analysis. Our empirical analysis is the first to demonstrate that price evolution is governed by a non-quadratic potential function. We find that the potential landscape exhibits scale-dependent topological transitions—from single-well to double-well structures—across timescales ranging from minutes to months: single wells characterize stable regimes, whereas double wells signal heightened uncertainty and systemic stress. Based on this, we propose a novel market-state classification criterion grounded in the topology of the inferred potential function, providing both a theoretical framework and empirical foundation for dynamic risk预警 and metastable regime identification.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Economics, Online Markets and Human Computation: Economic aspects of blockchain and cryptocurrencies
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This work builds upon the long-standing conjecture that linear diffusion models are inadequate for complex market dynamics. Specifically, it provides experimental validation for the author's prior arguments that realistic market dynamics are governed by higher-order (cubic and higher) non-linearities in the drift. As the diffusion drift is given by the negative gradient of a potential function, this means that a non-linear drift translates into a non-quadratic potential. These arguments were based both on general theoretical grounds as well as a structured approach to modeling the price dynamics which incorporates money flows and their impact on market prices. Here, we find direct confirmation of this view by analyzing high-frequency crypto currency data at different time scales ranging from minutes to months. We find that markets can be characterized by either a single-well or a double-well potential, depending on the time period and sampling frequency, where a double-well potential may signal market uncertainty or stress.

Problem

• Validating non-linear drift models in financial markets
• Analyzing high-frequency cryptocurrency data across time scales
• Characterizing market dynamics with single or double-well potentials

Innovation

• Non-linear drift modeling with cubic dynamics
• Double-well potential characterizing market uncertainty
• High-frequency cryptocurrency data validation across timescales