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.

翻译：本文探讨了Lévy过程的信息几何。我们首先推导了两个Lévy过程之间的$α$-散度，并以此为基础计算了与Lévy过程几何相关的Fisher信息矩阵及$α$-联络。此外，我们讨论了该信息几何在统计学中的应用。作为示例，我们研究了金融建模中多种Lévy过程的微分几何结构，包括修正稳定过程、CGMY模型以及方差伽玛过程。