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Limit theorems for stochastic processes

Author: Jean Jacod; Alʹbert Nikolaevich Shiri︠a︡ev
Publisher: Berlin ; New York : Springer, ©2003.
Series: Grundlehren der mathematischen Wissenschaften, 288.
Edition/Format:   Print book : English : 2nd edView all editions and formats
Summary:

This volume by two international leaders in the field proposes a systematic exposition of convergence in law for stochastic processes from the point of view of semimartingale theory.

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Jean Jacod; Alʹbert Nikolaevich Shiri︠a︡ev
ISBN: 3540439323 9783540439325
OCLC Number: 50554399
Description: xx, 660 pages ; 24 cm.
Contents: I. The general theory of stochastic processes, semimartingales and stochastic integrals --
1. Stochastic basis, stopping times, optional [sigma]-field, martingales --
2. Predictable [sigma]-field, predictable times --
3. Increasing processes --
4. Semimartingales and stochastic integrals --
II. Characteristics of semimartingales and processes with independent increments --
1. Random measures --
2. Characteristics of semimartingales --
3. Some examples --
4. Semimartingales with independent increments --
5. Processes with independent increments which are not semimartingales --
6. Process with conditionally independent increments --
7. Progressive conditional continuous PIIs --
8. Semimartingales, stochastic exponential and stochastic logarithm --
III. Martingale problems and changes of measures --
1. Martingale problems and point processes --
2. Martingale problems and semimartingales --
3. Absolutely continuous changes of measures --
4. Representation theorem for martingales --
5. Absolutely continuous change of measures: Explicit computation of the density process --
6. Integrals of vector-valued processes and [sigma]-martingales --
7. Laplace cumulant processes and Esscher's change of measures --
IV. Hellinger processes, absolute continuity and singularity of measures --
1. Hellinger integrals and Hellinger processes --
2. Predictable criteria for absolute continuity and singularity --
3. Hellinger processes for solutions of martingale problems --
4. Examples --
V. Contiguity, entire separation, convergence in variation --
1. Contiguity and entire separation --
2. Predictable criteria for contiguity and entire separation --
3. Examples --
4. Variation metric --
VI. Skorokhod topology and convergence of processes --
1. The Skorokhod topology --
2. Continuity for the Skorokhod topology --
3. Weak convergence --
4. Criteria for tightness: The quasi-left continuous case --
5. Criteria for tightness: The general case --
6. Convergence, quadratic variation, stochastic integrals --
VII. Convergence of processes with independent increments --
1. Introduction to functional limit theorems --
2. Finite-dimensional convergence --
3. Functional convergence and characteristics --
4. More on the general case --
5. The central limit theorem --
VIII. Convergence to a process with independent increments --
1. Finite-dimensional convergence, a general theorem --
2. Convergence to a PII without fixed time of discontinuity --
3. Applications --
4. Convergence to a general process with independent increments --
5. Convergence to a mixture of PII's, stable convergence and mixing convergence --
IX. Convergence to a semimartingale --
1. Limits of martingales --
2. Identification of the limit --
3. Limit theorems for semimartingales --
4. Applications --
5. Convergence of stochastic integrals --
6. Stability for stochastic differential equation --
7. Stable convergence to a progressive conditional continuous PII --
X. Limit theorems, density processes and contiguity --
1. Convergence of the density processes to a continuous process --
2. Convergence of the log-likelihood to a process with independent increments --
3. The statistical invariance principle.
Series Title: Grundlehren der mathematischen Wissenschaften, 288.
Responsibility: Jean Jacod, Albert N. Shiryaev.
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as a landmark in probability theory. The same can be said about the second edition. I can recommend this book to every reader who is sufficiently experienced and willing to spend some effort ... . Read more...

 
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