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Information geometry of Lévy processes and financial models

Jaehyung Choi

TL;DR

This work develops an information-geometric framework for Lévy processes by deriving the $α$-divergence between processes and using it to define the Fisher information matrix and the $α$-connection. It extends previous tempered-stable results to the full Lévy class, clarifying how the Brownian component $σ$ and the martingale condition affect the geometry. Through concrete financial-model examples—tempered stable, CGMY (CTS), and variance-gamma—the paper provides model-specific divergences and geometric structures, illustrating statistical benefits such as bias reduction and Bayesian predictive priors. The results offer practical tools for calibration, risk management, and robust inference in jump-diffusion finance by linking divergence-based geometry with stochastic process modeling.

Abstract

We explore the information geometry of Lévy processes. As a starting point, we derive the $α$-divergence between two Lévy processes. Subsequently, the Fisher information matrix and the $α$-connection associated with the geometry of Lévy processes are computed from the $α$-divergence. In addition, we discuss statistical applications of this information geometry. As illustrative examples, we investigate the differential-geometric structures of various Lévy processes relevant to financial modeling, including tempered stable processes, the CGMY model, and variance gamma processes.

Information geometry of Lévy processes and financial models

TL;DR

This work develops an information-geometric framework for Lévy processes by deriving the -divergence between processes and using it to define the Fisher information matrix and the -connection. It extends previous tempered-stable results to the full Lévy class, clarifying how the Brownian component and the martingale condition affect the geometry. Through concrete financial-model examples—tempered stable, CGMY (CTS), and variance-gamma—the paper provides model-specific divergences and geometric structures, illustrating statistical benefits such as bias reduction and Bayesian predictive priors. The results offer practical tools for calibration, risk management, and robust inference in jump-diffusion finance by linking divergence-based geometry with stochastic process modeling.

Abstract

We explore the information geometry of Lévy processes. As a starting point, we derive the -divergence between two Lévy processes. Subsequently, the Fisher information matrix and the -connection associated with the geometry of Lévy processes are computed from the -divergence. In addition, we discuss statistical applications of this information geometry. As illustrative examples, we investigate the differential-geometric structures of various Lévy processes relevant to financial modeling, including tempered stable processes, the CGMY model, and variance gamma processes.

Paper Structure

This paper contains 8 sections, 7 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Lévy processes
  3. Information geometry of Lévy processes
  4. Examples
  5. Tempered stable processes
  6. CTS processes and CGMY model
  7. Variance gamma processes
  8. Conclusion

Key Result

Proposition 1

Let $(X_t, \mathbb{P})_{t\in[0,T]}$ and $(X_t, \mathbb{Q})_{t\in[0,T]}$ be two Lévy processes on $(\Omega,\mathcal{F})$ with characteristic triplets $(\sigma_\mathbb{P},\nu_\mathbb{P},\gamma_{\mathbb{P}})$ and $(\sigma_\mathbb{Q},\nu_\mathbb{Q},\gamma_{\mathbb{Q}})$, respectively. Then $\mathbb{P}|_ The Radon--Nikodym derivative is given by where $(U_t,\mathbb{U})_{t\in[0,T]}$ is a Lévy process w

Theorems & Definitions (12)

  • Proposition 1: Sato (1999)
  • Proposition 2: Cont and Tankov (2004)
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • ...and 2 more