🤖 AI Summary
This paper systematically establishes an information-geometric framework for Lévy processes. **Problem:** Classical information geometry applies primarily to smooth, Gaussian, or Markovian families; extending it to non-Gaussian, non-Markovian jump processes—ubiquitous in finance—remains open. **Method:** We introduce the α-divergence for Lévy processes, from which we derive the Fisher information metric and α-connections, thereby endowing the parameter manifold with a differential-geometric structure. Our approach generalizes finite-dimensional information geometry to infinite-dimensional jump processes and yields explicit computations for tempered stable, CGMY, and variance gamma processes. **Contribution/Results:** (1) We present the first rigorous information-geometric formalism applicable to non-Gaussian, non-Markovian Lévy families; (2) we uncover intrinsic links between curvature of the parameter manifold and statistical asymptotics; and (3) we provide geometric tools for financial model selection, parameter estimation, and robustness analysis, advancing a geometric paradigm for statistical inference on stochastic processes.
📝 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.