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. It emphasizes results that are useful for mathematical theory and mathematical statistics. Coverage develops in detail useful parts of the general theory of stochastic processes, such as martingale problems and absolute continuity or contiguity results.
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Editorial Reviews
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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 Lépingle, 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 Back Cover
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. It should be useful to the professional probabilist or mathematical statistician, and of interest also to graduate students.
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This review is from: Limit Theorems for Stochastic Processes (Hardcover)
Just about every time I open this book I either find the elucidation of a concept which either I have always wanted to learn; or see the connection between ideas which I have known for some time.
This valuable work unifies a number of topics which are of great importance to the mathematical practitioner. Each of these is treated not merely as noetic nicety but as tool for applying the theory. The thorough and extensive treatment of continguity theory for point processes and convergence of stochastic integrals are especially well done and satisfying. Although even a two semester course does not suffice to cover the entire book I nevertheless feel that the dedicated educator should be able to delineate a number of threads for two one-semeter graduate courses.
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Inside This Book
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First Sentence:
The "General Theory of Stochastic Processes", in spite of its name, encompasses the rather restrictive subject of stochastic processes indexed by R+. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
strict stopping time, majoration condition, finite variation over finite intervals, majoration hypothesis, local uniform topology, evanescent set, strong majoration, modified second characteristic, triangular array scheme, very good extension, purely discontinuous local martingale, diffusions with jumps, symmetric nonnegative matrix, multivariate point processes, stochastic basis, cumulant processes, conditionally independent increments, smallest filtration, special semimartingale, announcing sequence, predictable random measure, filtered space, martingale problem, predictable criteria, bounded predictable processes Key Phrases - Capitalized Phrases (CAPs): (learn more)
Girsanov's Theorem, Progressive Conditional Continuous, Hence Theorem, The Statistical Invariance Principle, Therefore Theorem, Trotter-Kato's Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The "General Theory of Stochastic Processes", in spite of its name, encompasses the rather restrictive subject of stochastic processes indexed by R+. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
strict stopping time, majoration condition, finite variation over finite intervals, majoration hypothesis, local uniform topology, evanescent set, strong majoration, modified second characteristic, triangular array scheme, very good extension, purely discontinuous local martingale, diffusions with jumps, symmetric nonnegative matrix, multivariate point processes, stochastic basis, cumulant processes, conditionally independent increments, smallest filtration, special semimartingale, announcing sequence, predictable random measure, filtered space, martingale problem, predictable criteria, bounded predictable processes Key Phrases - Capitalized Phrases (CAPs): (learn more)
Girsanov's Theorem, Progressive Conditional Continuous, Hence Theorem, The Statistical Invariance Principle, Therefore Theorem, Trotter-Kato's Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 77
books:
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100
books
cite this book:
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- Complex Manifolds and Deformation of Complex Structures (Classics in Mathematics) by Kunihiko Kodaira in Back Matter
- Cohomology of Finite Groups (Grundlehren der mathematischen Wissenschaften) by Alejandro Adem in Back Matter
- Ordinary Differential Equations (Universitext) by Vladimir I. Arnold in Back Matter
- Sheaves on Manifolds: With a Short History "Les debuts de la theorie des faisceaux" by Christian Houzel (Grundlehren der mathematischen Wissenschaften) by Masaki Kashiwara in Back Matter
- Convex Analysis (Princeton Landmarks in Mathematics and Physics) by Ralph Tyrell Rockafellar in Back Matter
See all 77 books this book cites
- Semimartingales and their Statistical Inference (Chapman & Hall/CRC Monographs on Statistics & Applied Probability) by B.L.S. Prakasa Rao on 7 pages
- Levy Processes by Ole Eiler Barndorff-Nielsen on 5 pages
- Handbooks in Mathematical Finance: Option Pricing, Interest Rates and Risk Management by E. Jouini on page 40, and page 153
- Exotic Option Pricing and Advanced Lvy Models (Wilmott Collection) by Andreas Kyprianou on page 290, and page 304
- Probability Theory III: Stochastic Calculus (Encyclopaedia of Mathematical Sciences) (No. 3) by Yurij V. Prokhorov on page 155, and Back Matter
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