Limit Theorems for Stochastic Processes

By: Jean Jacod, Albert Shiryaev

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Hardcover | 10 October 2002 | Edition Number 2

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Hardcover
692 Pages

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Revised

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24.13 x 16.51 x 3.81

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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.

Industry Reviews

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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Non-Fiction Mathematics Probability & Statistics Applied Mathematics Stochastics

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Published: 18th December 2010

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The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals	p. 1
Stochastic Basis, Stopping Times, Optional ¿-Field, Martingales	p. 1
Stochastic Basis	p. 2
Stopping Times	p. 4
The Optional ¿-Field	p. 5
The Localization Procedure	p. 8
Martingales	p. 10
The Discrete Case	p. 13
Predictable ¿-Field, Predictable Times	p. 16
The Predictable ¿-Field	p. 16
Predictable Times	p. 17
Totally Inaccessible Stopping Times	p. 20
Predictable Projection	p. 22
The Discrete Case	p. 25
Increasing Processes	p. 27
Basic Properties	p. 27
Doob-Meyer Decomposition and Compensators of Increasing Processes	p. 32
Lenglart Domination Property	p. 35
The Discrete Case	p. 36
Semimartingales and Stochastic Integrals	p. 38
Locally Square-Integrable Martingales	p. 38
Decompositions of a Local Martingale	p. 40
Semimartingales	p. 43
Construction of the Stochastic Integral	p. 46
Quadratic Variation of a Semimartingale and Ito's Formula	p. 51
Doléans-Dade Exponential Formula	p. 58
The Discrete Case	p. 62
Characteristics of Semimartingales and Processes with Independent Increments	p. 64
Random Measures	p. 64
General Random Measures	p. 65
Integer-Valued Random Measures	p. 68
A Fundamental Example: Poisson Measures	p. 70
Stochastic Integral with Respect to a Random Measure	p. 71
Characteristics of Semimartingales	p. 75
Definition of the Characteristics	p. 75
Integrability and Characteristics	p. 81
A Canonical Representation for Semimartingales	p. 84
Characteristics and Exponential Formula	p. 85
Some Examples	p. 91
The Discrete Case	p. 91
More on the Discrete Case	p. 93
The "One-Point" Point Process and Empirical Processes	p. 97
Semimartingales with Independent Increments	p. 101
Wiener Processes	p. 102
Poisson Processes and Poisson Random Measures	p. 103
Processes with Independent Increments and Semimartingales	p. 106
Gaussian Martingales	p. 111
Processes with Independent Increments Which Are Not Semimartingales	p. 114
The Results	p. 114
The Proofs	p. 116
Processes with Conditionally Independent Increments	p. 124
Progressive Conditional Continuous PIIs	p. 128
Semimartingales, Stochastic Exponential and Stochastic Logarithm	p. 134
More About Stochastic Exponential and Stochastic Logarithm	p. 134
Multiplicative Decompositions and Exponentially Special Semimartingales	p. 138
Martingale Problems and Changes of Measures	p. 142
Martingale Problems and Point Processes	p. 143
General Martingale Problems	p. 143
Martingale Problems and Random Measures	p. 144
Point Processes and Multivariate Point Processes	p. 146
Martingale Problems and Semimartingales	p. 151
Formulation of the Problem	p. 152
Example: Processes with Independent Increments	p. 154
Diffusion Processes and Diffusion Processes with Jumps	p. 155
Local Uniqueness	p. 159
Absolutely Continuous Changes of Measures	p. 165
The Density Process	p. 165
Girsanov's Theorem for Local Martingales	p. 168
Girsanoy's Theorem for Random Measures	p. 170
Girsanov's Theorem for Semimartingales	p. 172
The Discrete Case	p. 177
Representation Theorem for Martingales	p. 179
Stochastic Integrals with Respect to a Multi-Dimensional Continuous Local Martingale	p. 179
Projection of a Local Martingale on a Random Measure	p. 182
The Representation Property	p. 185
The Fundamental Representation Theorem	p. 187
Absolutely Continuous Change of Measures: Explicit Computation of the Density Process	p. 191
All P-Martingales Have the Representation Property Relative to X	p. 192
P′ Has the Local Uniqueness Property	p. 196
Examples	p. 200
Integrals of Vector-Valued Processes and ¿-martingales	p. 203
Stochastic Integrals with Respect to a Multi-Dimensional Locally Square-integrable Martingale	p. 204
Integrals with Respect to a Multi-Dimensional Process of Locally Finite Variation	p. 206
Stochastic Integrals with Respect to a Multi-Dimensional Semimartingale	p. 207
Stochastic Integrals: A Predictable Criterion	p. 212
¿-localization and ¿-martingales	p. 214
Laplace Cumulant Processes and Esscher's Change of Measures	p. 219
Laplace Cumulant Processes of Exponentially Special Semimartingales	p. 219
Esscher Change of Measure	p. 222
Hellinger Processes, Absolute Continuity and Singularity of Measures	p. 227
Hellinger Integrals and Hellinger Processes	p. 228
Kakutani-Hellinger Distance and Hellinger Integrals	p. 228
Hellinger Processes	p. 230
Computation of Hellinger Processes in Terms of the Density Processes	p. 234
Some Other Processes of Interest	p. 237
The Discrete Case	p. 242
Predictable Criteria for Absolute Continuity and Singularity	p. 245
Statement of the Results	p. 245
The Proofs	p. 248
The Discrete Case	p. 252
Hellinger Processes for Solutions of Martingale Problems	p. 254
The General Setting	p. 255
The Case Where P and P′ Are Dominated by a Measure Having the Martingale Representation Property	p. 257
The Case Where Local Uniqueness Holds	p. 266
Examples	p. 272
Point Processes and Multivariate Point Processes	p. 272
Generalized Diffusion Processes	p. 275
Processes with Independent Increments	p. 277
Contiguity, Entire Separation, Convergence in Variation	p. 284
Contiguity and Entire Separation	p. 284
General Facts	p. 284
Contiguity and Filtrations	p. 290
Predictable Criteria for Contiguity and Entire Separation	p. 291
Statements of the Results	p. 291
The Proofs	p. 294
The Discrete Case	p. 301
Examples	p. 304
Point Processes	p. 304
Generalized Diffusion Processes	p. 305
Processes with Independent Increments	p. 306
Variation Metric	p. 309
Variation Metric and Hellinger Integrals	p. 310
Variation Metric and Hellinger Processes	p. 312
Examples: Point Processes and Multivariate Point Processes	p. 318
Example: Generalized Diffusion Processes	p. 322
Skorokhod Topology and Convergence of Processes	p. 324
The Skorokhod Topology	p. 325
Introduction and Notation	p. 325
The Skorokhod Topology: Definition and Main Results	p. 327
Proof of Theorem 1.14	p. 329
Continuity for the Skorokhod Topology	p. 337
Continuity Properties of some Functions	p. 337
Increasing Functions and the Skorokhod Topology	p. 342
Weak Convergence	p. 347
Weak Convergence of Probability Measures	p. 347
Application to Càdlàg Processes	p. 348
Criteria for Tightness: The Quasi-Left Continuous Case	p. 355
Aldous' Criterion for Tightness	p. 356
Application to Martingales and Semimartingales	p. 358
Criteria for Tightness: The General Case	p. 362
Criteria for Semimartingales	p. 362
An Auxiliary Result	p. 365
Proof of Theorem 5.17	p. 367
Convergence, Quadratic Variation, Stochastic Integrals	p. 376
The P-UT Condition	p. 377
Tightness and the P-UT Property	p. 382
Convergence of Stochastic Integrals and Quadratic Variation	p. 382
Some Additional Results	p. 386
Convergence of Processes with Independent Increments	p. 389
Introduction to Functional Limit Theorems	p. 390
Finite-Dimensional Convergence	p. 394
Convergence of Infinitely Divisible Distributions	p. 394
Some Lemmas on Characteristic Functions	p. 398
Convergence of Rowwise Independent Triangular Arrays	p. 402
Finite-Dimensional Convergence of PII-Semimartingales to a PII Without Fixed Time of Discontinuity	p. 408
Functional Convergence and Characteristics	p. 413
The Results	p. 414
Sufficient Condition for Convergence Under 2.48	p. 418
Necessary Condition for Convergence	p. 418
Sufficient Condition for Convergence	p. 424
More on the General Case	p. 428
Convergence of Non-Infinitesimal Rowwise Independent Arrays	p. 428
Finite-Dimensional Convergence for General PII	p. 436
Another Necessary and Sufficient Condition for Functional Convergence	p. 439
The Central Limit Theorem	p. 444
The Lindeberg-Feller Theorem	p. 445
Zolotarev's Type Theorems	p. 446
Finite-Dimensional Convergence of PII's to a Gaussian Martingale	p. 450
Functional Convergence of PII's to a Gaussian Martingale	p. 452
Convergence to a Process with Independent Increments	p. 456
Finite-Dimensional Convergence, a General Theorem	p. 456
Description of the Setting for This Chapter	p. 456
The Basic Theorem	p. 457
Remarks and Comments	p. 459
Convergence to a PII Without Fixed Time of Discontinuity	p. 460
Finite-Dimensional Convergence	p. 461
Functional Convergence	p. 464
Application to Triangular Arrays	p. 465
Other Conditions for Convergence	p. 467
Applications	p. 469
Central Limit Theorem: Necessary and Sufficient Conditions	p. 470
Central Limit Theorem: The Martingale Case	p. 473
Central Limit Theorem for Triangular Arrays	p. 477
Convergence of Point Processes	p. 478
Normed Sums of I.I.D. Semimartingales	p. 481
Limit Theorems for Functionals of Markov Processes	p. 486
Limit Theorems for Stationary Processes	p. 489
Convergence to a General Process with Independent Increments	p. 499
Proof of Theorem 4.1 When the Characteristic Function of X_t Vanishes Almost Nowhere	p. 501
Convergence of Point Processes	p. 503
Convergence to a Gaussian Martingale	p. 504
Convergence to a Mixture of PII's, Stable Convergence and Mixing Convergence	p. 506
Convergence to a Mixture of PII's	p. 506
More on the Convergence to a Mixture of PII's	p. 510
Stable Convergence	p. 512
Mixing Convergence	p. 518
Application to Stationary Processes	p. 519
Convergence to a Semimartingale	p. 521
Limits of Martingales	p. 521
The Bounded Case	p. 522
The Unbounded Case	p. 524
Identification of the Limit	p. 527
Introductory Remarks	p. 527
Identification of the Limit: The Main Result	p. 530
Identification of the Limit Via Convergence of the Characteristics	p. 533
Application: Existence of Solutions to Some Martingale Problems	p. 535
Limit Theorems for Semimartingales	p. 540
Tightness of the Sequence (Xⁿ)	p. 541
Limit Theorems: The Bounded Case	p. 546
Limit Theorems: The Locally Bounded Case	p. 550
Applications	p. 554
Convergence of Diffusion Processes with Jumps	p. 554
Convergence of Step Markov Processes to Diffusions	p. 557
Empirical Distributions and Brownian Bridge	p. 560
Convergence to a Continuous Semimartingale: Necessary and Sufficient Conditions	p. 561
Convergence of Stochastic Integrals	p. 564
Characteristics of Stochastic Integrals	p. 564
Statement of the Results	p. 567
The Proofs	p. 570
Stability for Stochastic Differential Equation	p. 575
Auxiliary Results	p. 576
Stochastic Differential Equations	p. 577
Stability	p. 578
Table of Contents provided by Publisher. All Rights Reserved.

Limit Theorems for Stochastic Processes

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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Paperback

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