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

翻译：我们发展了Lévy过程的信息几何。通过直接以Lévy过程的三元组表示α-散度，我们识别了统计流形上的Fisher信息矩阵与α-联络。此外，我们探讨了该信息几何的统计学意义，包括偏差缩减估计与贝叶斯预测先验。以金融建模中广泛使用的若干Lévy过程（如温稳态过程、CGMY模型、方差伽马过程及默顿模型）为例，通过其微分几何结构进行了研究。