Limit Theorems for Stochastic Processes

Front Cover
Springer Science & Business Media, Mar 9, 2013 - Mathematics - 664 pages
Initially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is only recently that the relation between the two theories has been thoroughly studied. The authors of this Grundlehren volume, two of the international leaders in the field, propose a systematic exposition of convergence in law for stochastic processes, from the point of view of semimartingale theory, with emphasis on results that are useful for mathematical theory and mathematical statistics. This leads them to develop in detail some particularly useful parts of the general theory of stochastic processes, such as martingale problems, and absolute continuity or contiguity results. The book contains an introduction to the theory of martingales and semimartingales, random measures stochastic integrales, Skorokhod topology, etc., as well as a large number of results which have never appeared in book form, and some entirely new results. The second edition contains some additions to the text and references. Some parts are completely rewritten.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

The General Theory of Stochastic Processes
1
Convergence to a Mixture of PIIs Stable Convergence
5
Predictable oField Predictable Times
16
Increasing Processes
27
Semimartingales and Stochastic Integrals
38
Characteristics of Semimartingales and Processes
64
Characteristics of Semimartingales
75
Some Examples
91
Weak Convergence
347
The QuasiLeft Continuous Case
355
The General Case
362
Convergence Quadratic Variation Stochastic Integrals
376
Convergence of Processes with Independent Increments
389
Functional Convergence and Characteristics
413
More on the General Case
428
The Central Limit Theorem
444

Semimartingales with Independent Increments
101
Processes with Independent Increments
112
Processes with Conditionally Independent Increments
124
Semimartingales Stochastic Exponential and Stochastic Logarithm
134
Martingale Problems and Changes of Measures
142
Martingale Problems and Semimartingales
151
Absolutely Continuous Changes of Measures
165
Representation Theorem for Martingales
179
Integrals of VectorValued Processes and omartingales
203
Laplace Cumulant Processes and Esschers Change of Measures
219
Hellinger Processes Absolute Continuity
227
Predictable Criteria for Absolute Continuity and Singularity
245
Hellinger Processes for Solutions of Martingale Problems
254
3c The Case Where Local Uniqueness Holds
266
Examples
272
Contiguity Entire Separation Convergence in Variation
284
Predictable Criteria for Contiguity and Entire Separation
291
Examples
304
Skorokhod Topology and Convergence of Processes
324
Continuity for the Skorokhod Topology
337
5d Functional Convergence of PIIs to a Gaussian Martingale
452
Convergence to a PII Without Fixed Time of Discontinuity
460
Applications
469
Convergence to a General Process with Independent Increments
499
Convergence to a Semimartingale
521
Identification of the Limit
527
Limit Theorems for Semimartingales
540
Applications
554
Convergence of Stochastic Integrals
564
Stability for Stochastic Differential Equation
575
Stable Convergence to a Progressive Conditional Continuous PII
583
Limit Theorems Density Processes and Contiguity
592
Convergence of the LogLikelihood to a Process
612
The Statistical Invariance Principle
620
Bibliographical Comments
629
References
641
Index of Symbols 653
652
Index of Topics
659
Copyright

Other editions - View all

Common terms and phrases

Bibliographic information