Limit Theorems for Stochastic Processes

Jean Jacod, Albert N. Shiryaev (auth.)9783540439325, 3540439323

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 integrals, 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.

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Table of contents :
Front Matter….Pages I-XVII
The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals….Pages 1-63
Characteristics of Semimartingales and Processes with Independent Increments….Pages 64-128
Martingale Problems and Changes of Measures….Pages 129-190
Hellinger Processes, Absolute Continuity and Singularity of Measures….Pages 191-247
Contiguity, Entire Separation, Convergence in Variation….Pages 248-287
Skorokhod Topology and Convergence of Processes….Pages 288-347
Convergence of Processes with Independent Increments….Pages 348-414
Convergence to a Process with Independent Increments….Pages 415-479
Convergence to a Semimartingale….Pages 480-534
Limit Theorems, Density Processes and Contiguity….Pages 535-571
Back Matter….Pages 572-604