Information geometry of L\'evy processes and financial models

TL;DR

This paper explores the geometric structure of Le9vy processes used in finance, deriving divergence measures and geometric tools to improve statistical inference and modeling accuracy.

Contribution

It introduces the information geometry framework for Le9vy processes, deriving divergence measures and geometric structures directly from their triplets, with applications to financial models.

Findings

Derived b1-divergences from Le9vy triplets

Identified Fisher information and b1-connection on the manifold

Analyzed geometric structures of key financial Le9vy models

Abstract

We develop the information geometry of L\'evy processes. Deriving $\alpha$-divergences directly in terms of the L\'evy triplets of the L\'evy processes, we identify Fisher information matrix and $\alpha$-connection on the statistical manifold. In addition, we discuss statistical implications of this information geometry, including bias reduction estimation and Bayesian predictive priors. Several L\'evy 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.