本书是一部全新、综合描述Lévy过程理论的教程。近年来，Lévy过程理论作为现代概率的重要一支得到了迅速的发展，在序列、数学金融和风险估计等各个领域的应用广泛。Bertoin教授运用概率结构和分析工具之间强有力的联系将这个核心理论讲述的相当简明。介绍从属过程的特殊性质以及其在研究实值Lévy过程和起伏理论时的关键特征。详尽讲述了没有正跳跃的Lévy过程和平稳过程。

Preface

O Preliminaries

 1 Notation

 2 Infinitely divisible distributions

 3 Martingales

 4 Poisson processes

 5 Poisson measures and Poisson point processes

 6 Brownian motion

 7 Regular variation and Tauberian theorems

Ⅰ Lévy Processes as Markov Processes

 1 Lévy processes and the Lbvy-Khintchine formula

 2 Markov property and related operators

 3 Absolutely continuous resoivents

 4 Transience and recurrence

 5 Exercises

 6 Comments

Ⅱ Elements of Potential Theory

 1 Duality and time reversal

 2 Capacitary measure

 3 Essentially polar sets and capacity

 4 Energy

 5 The case of a single point

 6 Exercises

 7 Comments

Ⅲ Subordimtors

 1 Definitions and first properties

 2 Passage across a level

 3 The arcsine laws

 4 Rates of growth

 5 Dimension of the range

 6 Exercises

 7 Comments

Ⅳ Local Time and Excursions of a Markov Process

 1 Framework

 2 Construction of the local time

 3 Inverse local time

 4 Excursion measure and excursion process

 5 The cases of holding points and of irregular points

 6 Exercises

 7 Comments

Ⅴ Local Times of a Lévy Process

 1 Occupation measure and local times

 2 Hilbert transform of local times

 3 Jointly continuous local times

 4 Exercises

 5 Comments

Ⅵ Fluctuation Theory

 1 The reflected process and the ladder process

 2 Fluctuation identities

 3 Some applications of the ladder time process

 4 Some applications of the ladder height process

 5 Increase times

 6 Exercises

 7 Comments

Ⅶ Lévy Processes with no Positive Jumps

 1 Fluctuation theory with no positive jumps

 2 The scale function

 3 The process conditioned to stay positive

 4 Some path transformations

 5 Exercises

 6 Comments

Ⅷ Stable Processes and the Scaling Property

 1 Definition and probability estimates

 2 Some sample path properties

 3 Bridges

 4 Normalized excursion and meander

 5 Exercises

 6 Comments

References

List of symbols

Index