Paper: arXiv 2509.02941

Authors: Igor Halperin

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.

Complexity vs Empirical Score

  • Math Complexity: 0.0/10
  • Empirical Rigor: 0.0/10
  • Quadrant: Philosophers — conceptual discussion, limited math and data

Why this score: No content returned from model

Research Flowchart

  flowchart TD
  A["Research Goal<br>Validate non-linear drift in<br>market price dynamics"] --> B["Methodology<br>High-Frequency Crypto Data Analysis"]
  B --> C["Input Data<br>High-frequency BTC/USD<br>tick data"]
  C --> D["Computational Process<br>Estimate Stochastic Differential Equation<br>Drift & Diffusion Coefficients"]
  D --> E["Computational Process<br>Construct Potential Function<br>U(x) via integration"]
  E --> F{"Findings<br>Market Potential Characterization"}
  F --> G["Single-well Potential<br>Periods of stability"]
  F --> H["Double-well Potential<br>Signals market uncertainty/stress"]