Information geometry of Lévy processes and financial models

Paper: arXiv 2507.23646

Authors: Jaehyung Choi

Abstract

We develop the information geometry of Lévy processes. Deriving $α$-divergences directly in terms of the Lévy triplets of the Lévy processes, we identify Fisher information matrix and $α$-connection on the statistical manifold. In addition, we discuss statistical implications of this information geometry, including bias reduction estimation and Bayesian predictive priors. Several Lévy processes, broadly used for financial modeling such as tempered stable processes, the CGMY model, variance gamma processes, and the Merton model, are investigated through their differential-geometric structures as illustrative examples.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical generalization of information geometry to the full class of Lévy processes, deriving explicit expressions for alpha-divergences, Fisher information matrices, and alpha-connections. However, it is purely theoretical with no empirical backtests or empirical data analysis presented in the provided excerpt.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Develop Information Geometry of Lévy Processes"] --> B["Inputs:<br>Lévy Triplets of Processes<br><i>(Tempered Stable, CGMY, VG, Merton)</i>"]
    B --> C["Derive α-Divergences<br>directly from Lévy Triplets"]
    C --> D["Identify Differential-Geometric Structure:<br>Fisher Information Matrix & α-Connection"]
    D --> E["Statistical Applications:<br>Bias Reduction Estimation & Bayesian Predictive Priors"]
    E --> F["Outcomes:<br>Geometric characterization & enhanced statistical inference for financial models"]