随机过程用的极限定理 : 第2版

Chapter Ⅰ The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals

1.Stochastic Basis, Stopping Times, Optional a -Field, Martingales

a.Stochastic Basis

b.Stopping Times

c.The Optional σ-Field

d.The Localization Procedure

e.Martingales

f.The Discrete Case

2.Predictable σ-Field, Predictable Times

a.The Predictable σ-Field

b.Predictable Times

c.Totally Inaccessible Stopping Times

d.Predictable Projection

e.The Discrete Case

3.Increasing Processes

a.Basic Properties

b.Doob-Meyer Decomposition and Compensators of Increasing Processes

c.Lenglart Domination Property

d.The Discrete Case

4.Semimartingales and Stochastic Integrals

a.Locally Square-Integrable Martingales

b.Decompositions of a Local Martingale

c.Semimartingales

d.Construction of the Stochastic Integral

e.Quadratic Variation of a Semimartingale and Ito's Formula

f.Doleans-Dade Exponential Formula

g.The Discrete Case

Chapter Ⅱ Characteristics of Semimartingales and Processes with Independent Increments

1.Random Measures

1a.General Random Measures

1b.Integer-Valued Random Measures

1c.A Fundamental Example: Poisson Measures

1d.Stochastic Integral with Respect to a Random Measure

2.Characteristics of Semimartingales

2a.Definition of the Characteristics

2b.Integrability and Characteristics

2c.A Canonical Representation for Semimartingales

2d.Characteristics and Exponential Formula

3.Some Examples

3a.The Discrete Case

3b.More on the Discrete Case

3c.The "One-Point" Point Process and Empirical Processes

4.Semimartingales with Independent Increments

4a.Wiener Processes

4b.Poisson Processes and Poisson Random Measures

4c.Processes with Independent Increments and Semimartingales

4d.Gaussian Martingales

5.Processes with Independent Increments Which Are Not Semimartingales

5a.The Results

5b.The Proofs

6.Processes with Conditionally Independent Increments

7.Progressive Conditional Continuous PIIs

8.Semimartingales, Stochastic Exponential and Stochastic Logarithm

8a.More About Stochastic Exponential and Stochastic Logarithm,

8b.Multiplicative Decompositions and Exponentially Special Semimartingales

Chapter Ⅲ Martingale Problems and Changes of Measures

1.Martingale Problems and Point Processes

1a.General Martingale Problems

1b.Martingale Problems and Random Measures

1c.Point Processes and Multivariate Point Processes

……

Chapter Ⅳ Bellinger Processes, Absolute Continuity

Chapter Ⅴ Contiguity, Entire Separation, Convergence in Variation

Chapter Ⅵ Skorokhod Topology and Convergence of Processes

Chapter Ⅶ Convergence of Processes with Independent Increments

Chapter Ⅷ Convergence to a Process with Independent Increments

Chapter Ⅸ Convergence to a Semimartingale

Chapter Ⅹ Limit Theorems, Density Processes and Contiguity Bibliographical Comments

References

Index of Symbols

Index of Terminology

Index of Topics

Index of Conditions for Limit Theorems

Apart from correcting a number of printing mistakes， and some mathematical inaccuracies as well， this second edition contains some new material: indeed， during the fifteen years elapsed since the first edition came out， a large number of new results concerning limit theorems have of course been proved by many authors， and more generally mathematical life has been going on. This gave us the feeling that some of the material in the first edition was perhaps not as important as we thought at the time， while there were some neglected topics which have in fact proved to be very useful in various applications.
　　So perhaps a totally new book would have been a good thing to write. Our natural laziness prevented us to do that， but we have felt compelled to fill in the most evident holes in this book. This has been done in the most painless way for us， and also for the reader acquainted with the first edition （at least we hope so ...）. That is all new material has been added at the end of preexisting chapters.