Information geometry of Lévy processes and financial models

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.