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Encyclopedia of Mathematics and its Applications : Stochastic Integration with Jumps Series Number 89 - Klaus Bichteler

Encyclopedia of Mathematics and its Applications

Stochastic Integration with Jumps Series Number 89

Hardcover Published: 13th May 2002
ISBN: 9780521811293
Number Of Pages: 516

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Stochastic processes with jumps and random measures are gaining importance as drivers in applications like financial mathematics and signal processing. This book develops stochastic integration theory for both integrators (semimartingales) and random measures from a common point of view. Using some novel predictable controlling devices, the author furnishes the theory of stochastic differential equations driven by them, as well as their stability and numerical approximation theories. Highlights feature DCT and Egoroff's Theorem, as well as comprehensive analogs to results from ordinary integration theory, for instance previsible envelopes and an algorithm computing stochastic integrals of caglad integrands pathwise. Full proofs are given for all results, and motivation is stressed throughout. A large appendix contains most of the analysis that readers will need as a prerequisite. This will be an invaluable reference for graduate students and researchers in mathematics, physics, electrical engineering and finance who need to use stochastic differential equations.

'The material in the book is presented well: it is detailed, motivation is stressed throughout and the text is written with an enjoyable pinch of dry humour.' Evelyn Buckwar, Zentralblatt MATH 'The highlights of the monograph are: Girsanov-Meyer theory on shifted martingales, which covers both the Wiener and Poisson setting; a Doob-Meyer decomposition statement providing really deep information that the objects that can go through the Daniell-like construction of the stochastic. This is an excellent and informative monograph for a general mathematical audience.' EMS

Prefacep. xi
Introductionp. 1
Motivation: Stochastic Differential Equationsp. 1
Wiener Processp. 9
The General Modelp. 20
Integrators and Martingalesp. 43
The Elementary Stochastic Integralp. 46
The Semivariationsp. 53
Path Regularity of Integratorsp. 58
Processes of Finite Variationp. 67
Martingalesp. 71
Extension of the Integralp. 87
The Daniell Meanp. 88
The Integration Theory of a Meanp. 94
Countable Additivity in p-Meanp. 106
Measurabilityp. 110
Predictable and Previsible Processesp. 115
Special Properties of Daniell's Meanp. 123
The Indefinite Integralp. 130
Functions of Integratorsp. 145
Ito's Formulap. 157
Random Measuresp. 171
Control of Integral and Integratorp. 187
Change of Measure--Factorizationp. 187
Martingale Inequalitiesp. 209
The Doob-Meyer Decompositionp. 221
Semimartingalesp. 232
Previsible Control of Integratorsp. 238
Levy Processesp. 253
Stochastic Differential Equationsp. 271
Introductionp. 271
Existence and Uniqueness of the Solutionp. 282
Stability: Differentiability in Parametersp. 298
Pathwise Computation of the Solutionp. 310
Weak Solutionsp. 330
Stochastic Flowsp. 343
Semigroups, Markov Processes, and PDEp. 351
Complements to Topology and Measure Theoryp. 363
Notations and Conventionsp. 363
Topological Miscellaneap. 366
Measure and Integrationp. 391
Weak Convergence of Measuresp. 421
Analytic Sets and Capacityp. 432
Suslin Spaces and Tightness of Measuresp. 440
The Skorohod Topologyp. 443
The L[superscript p]-Spacesp. 448
Semigroups of Operatorsp. 463
Answers to Selected Problemsp. 470
Referencesp. 477
Index of Notationsp. 483
Indexp. 489
Full Indexes
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521811293
ISBN-10: 0521811295
Series: Encyclopedia of Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 516
Published: 13th May 2002
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.4 x 15.6  x 2.9
Weight (kg): 0.9