Information geometry of L\'evy processes and financial 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.