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商品簡介
本書是一部全新、綜合描述Levy過程理論的教程。近年來，Levy過程理論作為現代概率的重要一支得到了迅速的發展，在序列、數學金融和風險估計等各個領域的應用廣泛。Bertoin教授運用概率結構和分析工具之間強有力的聯系將這個核心理論講述的相當簡明。介紹從屬過程的特殊性質以及其在研究實值Levy過程和起伏理論時的關鍵特徵。詳盡講述了沒有正跳躍的Levy過程和平穩過程。目次：基礎；馬爾科夫過程的Levy過程；勢理論基礎；局部時間和馬爾科夫游弋；Levy過程的局部時間；起伏理論；沒有正跳躍的Levy過程；平穩過程和標度特徵。
讀者對象：本書適用于所有對概率論感興趣的科研人員。
讀者對象：本書適用于所有對概率論感興趣的科研人員。
目次
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
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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