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本书是一部全新、综合描述Levy过程理论的教程。近年来，Levy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Levy过程和起伏理论时的关键特征。详尽讲述了没有正跳跃的Levy过程和平稳过程。目次：基础；马尔科夫过程的Levy过程；势理论基础；局部时间和马尔科夫游弋；Levy过程的局部时间；起伏理论；没有正跳跃的Levy过程；平稳过程和标度特征。

读者对象：本书适用于所有对概率论感兴趣的科研人员。

Preface

0 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 Levy Processes as Markov Processes

1 Levy 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 Levy 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

Vll Levy 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

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书籍介绍

《Lévy过程》是一部全新、综合描述Levy过程理论的教程。近年来，Levy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Levy过程和起伏理论时的关键特征。详尽讲述了没有正跳跃的Levy过程和平稳过程。目次：基础；马尔科夫过程的Levy过程；势理论基础；局部时间和马尔科夫游弋；Levy过程的局部时间；起伏理论；没有正跳跃的Levy过程；平稳过程和标度特征。读者对象：《Lévy过程》适用于所有对概率论感兴趣的科研人员。