Overview

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.

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From the reviews of the second edition: This is the second edition of a well-known book. Fifteen years ago, the first edition ... proved essential for all people interested in functional convergence of stochastic processes. ... Some new materials have been included in the present edition. ... There is also an up-to-date account on predictable uniform tightness because there has been significant progress in this field since the first edition. No doubt that this book will continue being a necessary companion for stochasticians. (Dominique Lepingle, Mathematical Reviews, 2003 j) This is the second edition of the fundamental monograph ... . This new edition has grown by about 50 pages ... . These extensions make the book even more valuable and comprehensive for people working in mathematical finance, numerics of stochastic processes and, of course, statistics of stochastic processes. (Markus Reiss, Zentralblatt MATH, Vol. 1018, 2003) The 1987 version of the book was 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 ... . Also, I think the book is very useful as a reference. ... To conclude, this book is still the reference in this domain and as such I can definitely recommend it to both pure and applied probabilists who are interested in this topic. (A.P. Zwart, Nieuw Archief voor Wiskunde, Vol. 7 (2), 2006)

From the reviews of the second edition: <p> This is the second edition of a well-known book. Fifteen years ago, the first edition a ] proved essential for all people interested in functional convergence of stochastic processes. a ] Some new materials have been included in the present edition. a ] There is also an up-to-date account on predictable uniform tightness because there has been significant progress in this field since the first edition. No doubt that this book will continue being a necessary companion for stochasticians. (Dominique LA(c)pingle, Mathematical Reviews, 2003 j) <p> This is the second edition of the fundamental monograph a ] . This new edition has grown by about 50 pages a ] . These extensions make the book even more valuable and comprehensive for people working in mathematical finance, numerics of stochastic processes and, of course, statistics of stochastic processes. (Markus ReiA, Zentralblatt MATH, Vol. 1018, 2003) <p> The 1987 version of the book was 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 a ] . Also, I think the book is very useful as a reference. a ] To conclude, this book is still the reference in this domain and as such I can definitely recommend it to both pure and applied probabilists who are interested in this topic. (A.P. Zwart, Nieuw Archief voor Wiskunde, Vol. 7 (2), 2006)

From the reviews of the second edition: This is the second edition of a well-known book. Fifteen years ago, the first edition ... proved essential for all people interested in functional convergence of stochastic processes. ... Some new materials have been included in the present edition. ... There is also an up-to-date account on predictable uniform tightness because there has been significant progress in this field since the first edition. No doubt that this book will continue being a necessary companion for stochasticians. (Dominique Lepingle, Mathematical Reviews, 2003 j) This is the second edition of the fundamental monograph ... . This new edition has grown by about 50 pages ... . These extensions make the book even more valuable and comprehensive for people working in mathematical finance, numerics of stochastic processes and, of course, statistics of stochastic processes. (Markus Reiss, Zentralblatt MATH, Vol. 1018, 2003) The 1987 version of the book was 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 ... . Also, I think the book is very useful as a reference. ... To conclude, this book is still the reference in this domain and as such I can definitely recommend it to both pure and applied probabilists who are interested in this topic. (A.P. Zwart, Nieuw Archief voor Wiskunde, Vol. 7 (2), 2006)

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