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Analysis and Approximation of Rare Events: Representations and Weak Convergence Methods 2019 ed. [Kõva köide]

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  • Formaat: Hardback, 574 pages, kõrgus x laius: 235x155 mm, kaal: 1051 g, 1 Illustrations, color; 13 Illustrations, black and white; XIX, 574 p. 14 illus., 1 illus. in color., 1 Hardback
  • Sari: Probability Theory and Stochastic Modelling 94
  • Ilmumisaeg: 11-Aug-2019
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 1493995774
  • ISBN-13: 9781493995776
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Analysis and Approximation of Rare Events: Representations and Weak Convergence Methods 2019 ed.Suurem pilt
  • Formaat: Hardback, 574 pages, kõrgus x laius: 235x155 mm, kaal: 1051 g, 1 Illustrations, color; 13 Illustrations, black and white; XIX, 574 p. 14 illus., 1 illus. in color., 1 Hardback
  • Sari: Probability Theory and Stochastic Modelling 94
  • Ilmumisaeg: 11-Aug-2019
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 1493995774
  • ISBN-13: 9781493995776
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781493995776.html

This book presents broadly applicable methods for the large deviation and moderate deviation analysis of discrete and continuous time stochastic systems. A feature of the book is the systematic use of variational representations for quantities of interest such as normalized logarithms of probabilities and expected values.  By characterizing a large deviation principle in terms of Laplace asymptotics, one converts the proof of large deviation limits into the convergence of variational representations. These features are illustrated though their application to a broad range of discrete and continuous time models, including stochastic partial differential equations, processes with discontinuous statistics, occupancy models, and many others. The tools used in the large deviation analysis also turn out to be useful in understanding Monte Carlo schemes for the numerical approximation of the same probabilities and expected values. This connection is illustrated through the design and analysis of importance sampling and splitting schemes for rare event estimation.  The book assumes a solid background in weak convergence of probability measures and stochastic analysis, and is suitable for advanced graduate students, postdocs and researchers.


Arvustused

The book is very well organized and the structure of each chapter is helpful: notation, assumptions, statements, examples, proofs and comments are clearly separated. this makes the book a good reference for researchers interested in rare event analysis and approximation. (Charles-Edouard Bréhier, Mathematical Reviews, August, 2020) The current book requires a solid background in weak convergence of probability measures and stochastic analysis, and it is intended for advanced graduate students, post-doctoral fellows and researchers working in this area. (Anatoliy Swishchuk, zbMATH 1427.60003, 2020)

Part I Laplace Principle, Relative Entropy, and Elementary Examples
1 General Theory
3(28)
1.1 Large Deviation Principle
3(4)
1.2 An Equivalent Formulation of the Large Deviation Principle
7(13)
1.3 Basic Results in the Theory
20(9)
1.4 Notes
29(2)
2 Relative Entropy and Tightness of Measures
31(18)
2.1 Properties of Relative Entropy
31(13)
2.2 Tightness of Probability Measures
44(3)
2.3 Notes
47(2)
3 Examples of Representations and Their Application
49(30)
3.1 Representation for an IID Sequence
49(11)
3.1.1 Sanov's and Cramer's Theorems
51(1)
3.1.2 Tightness and Weak Convergence
52(1)
3.1.3 Laplace Upper Bound
53(1)
3.1.4 Laplace Lower Bound
54(1)
3.1.5 Proof of Lemma 3.5 and Remarks on the Proof of Sanov's Theorem
54(2)
3.1.6 Cramer's Theorem
56(4)
3.2 Representation for Functionals of Brownian Motion
60(9)
3.2.1 Large Deviation Theory of Small Noise Diffusions
63(2)
3.2.2 Tightness and Weak Convergence
65(1)
3.2.3 Laplace Upper Bound
66(1)
3.2.4 Compactness of Level Sets
67(1)
3.2.5 Laplace Lower Bound
68(1)
3.3 Representation for Functionals of a Poisson Process
69(6)
3.4 Notes
75(4)
Part II Discrete Time Processes
4 Recursive Markov Systems with Small Noise
79(40)
4.1 Process Model
79(2)
4.2 The Representation
81(2)
4.3 Form of the Rate Function
83(1)
4.4 Statement of the LDP
84(2)
4.5 Laplace Upper Bound
86(6)
4.5.1 Tightness and Uniform Integrability
86(2)
4.5.2 Weak Convergence
88(2)
4.5.3 Completion of the Laplace Upper Bound
90(1)
4.5.4 I is a Rate Function
91(1)
4.6 Properties of L(x, β)
92(6)
4.7 Laplace Lower Bound Under Condition 4.7
98(4)
4.7.1 Construction of a Nearly Optimal Control
99(1)
4.7.2 Completion of the Proof of the Laplace Lower Bound
99(1)
4.7.3 Approximation by Bounded Velocity Paths
100(2)
4.8 Laplace Lower Bound Under Condition 4.8
102(15)
4.8.1 Mollification
102(2)
4.8.2 Variational Bound for the Mollified Process
104(2)
4.8.3 Perturbation of L and Its Properties
106(2)
4.8.4 A Nearly Optimal Trajectory and Associated Control Sequence
108(4)
4.8.5 Tightness and Convergence of Controlled Processes
112(2)
4.8.6 Completion of the Proof of the Laplace Lower Bound
114(3)
4.9 Notes
117(2)
5 Moderate Deviations for Recursive Markov Systems
119(32)
5.1 Assumptions, Notation, and Theorem Statement
121(3)
5.2 The Representation
124(1)
5.3 Tightness and Limits for Controlled Processes
125(16)
5.3.1 Tightness and Uniform Integrability
125(4)
5.3.2 Identification of Limits
129(12)
5.4 Laplace Upper Bound
141(1)
5.5 Laplace Lower Bound
142(7)
5.6 Notes
149(2)
6 Empirical Measure of a Markov Chain
151(30)
6.1 Applications
151(2)
6.1.1 Markov Chain Monte Carlo
152(1)
6.1.2 Markov Modulated Dynamics
152(1)
6.2 The Representation
153(1)
6.3 Form of the Rate Function
154(2)
6.4 Assumptions and Statement of the LDP
156(2)
6.5 Properties of the Rate Function
158(2)
6.6 Tightness and Weak Convergence
160(2)
6.7 Laplace Upper Bound
162(1)
6.8 Laplace Lower Bound
163(10)
6.9 Uniform Laplace Principle
173(1)
6.10 Noncompact State Space
174(4)
6.11 Notes
178(3)
7 Models with Special Features
181(30)
7.1 Introduction
181(1)
7.2 Occupancy Models
182(21)
7.2.1 Preliminaries and Main Result
183(5)
7.2.2 Laplace Upper Bound
188(2)
7.2.3 Properties of the Rate Function
190(6)
7.2.4 Laplace Lower Bound
196(2)
7.2.5 Solution to Calculus of Variations Problems
198(5)
7.3 Two Scale Recursive Markov Systems with Small Noise
203(3)
7.3.1 Model and Assumptions
204(1)
7.3.2 Rate Function and the LDP
205(1)
7.3.3 Extensions
206(1)
7.4 Notes
206(5)
Part III Continuous Time Processes
8 Representations for Continuous Time Processes
211(34)
8.1 Representation for Infinite Dimensional Brownian Motion
212(13)
8.1.1 The Representation
212(2)
8.1.2 Preparatory Results
214(3)
8.1.3 Proof of the Upper Bound in the Representation
217(1)
8.1.4 Proof of the Lower Bound in the Representation
218(4)
8.1.5 Representation with Respect to a General Filtration
222(3)
8.2 Representation for Poisson Random Measure
225(17)
8.2.1 The Representation
225(4)
8.2.2 Preparatory Results
229(3)
8.2.3 Proof of the Upper Bound in the Representation
232(3)
8.2.4 Proof of the Lower Bound in the Representation
235(3)
8.2.5 Construction of Equivalent Controls
238(4)
8.3 Representation for Functionals of PRM and Brownian Motion
242(1)
8.4 Notes
243(2)
9 Abstract Sufficient Conditions for Large and Moderate Deviations in the Small Noise Limit
245(16)
9.1 Definitions and Notation
246(1)
9.2 Abstract Sufficient Conditions for LDP and MDP
247(6)
9.2.1 An Abstract Large Deviation Result
248(2)
9.2.2 An Abstract Moderate Deviation Result
250(3)
9.3 Proof of the Large Deviation Principle
253(2)
9.4 Proof of the Moderate Deviation Principle
255(4)
9.5 Notes
259(2)
10 Large and Moderate Deviations for Finite Dimensional Systems
261(34)
10.1 Small Noise Jump-Diffusion
262(1)
10.2 An LDP for Small Noise Jump-Diffusions
263(15)
10.2.1 Proof of the Large Deviation Principle
270(8)
10.3 An MDP for Small Noise Jump-Diffusions
278(15)
10.3.1 Some Preparatory Results
280(8)
10.3.2 Proof of the Moderate Deviation Principle
288(3)
10.3.3 Equivalence of Two Rate Functions
291(2)
10.4 Notes
293(2)
11 Systems Driven by an Infinite Dimensional Brownian Noise
295(24)
11.1 Formulations of Infinite Dimensional Brownian Motion
296(6)
11.1.1 The Representations
300(2)
11.2 General Sufficient Condition for an LDP
302(4)
11.3 Reaction--Diffusion SPDE
306(11)
11.3.1 The Large Deviation Theorem
306(5)
11.3.2 Qualitative Properties of Controlled Stochastic Reaction--Diffusion Equations
311(6)
11.4 Notes
317(2)
12 Stochastic Flows of Diffeomorphisms and Image Matching
319(24)
12.1 Notation and Definitions
321(3)
12.2 Statement of the LDP
324(4)
12.3 Weak Convergence for Controlled Flows
328(8)
12.4 Application to Image Analysis
336(6)
12.5 Notes
342(1)
13 Models with Special Features
343(40)
13.1 Introduction
343(1)
13.2 A Model with Discontinuous Statistics-Weighted Serve-the-Longest Queue
344(21)
13.2.1 Problem Formulation
345(2)
13.2.2 Form of the Rate Function and Statement of the Laplace Principle
347(4)
13.2.3 Laplace Upper Bound
351(2)
13.2.4 Properties of the Rate Function
353(2)
13.2.5 Laplace Lower Bound
355(10)
13.3 A Class of Pure Jump Markov Processes
365(15)
13.3.1 Large Deviation Principle
366(6)
13.3.2 Moderate Deviation Principle
372(8)
13.4 Notes
380(3)
Part IV Accelerated Monte Carlo for Rare Events
14 Rare Event Monte Carlo and Importance Sampling
383(30)
14.1 Example of a Quantity to be Estimated
383(4)
14.1.1 Relative Error
385(2)
14.2 Importance Sampling
387(9)
14.2.1 Importance Sampling for Rare Events
388(2)
14.2.2 Controls Without Feedback, and Dangers in the Rare Event Setting
390(3)
14.2.3 A Dynamic Game Interpretation of Importance Sampling
393(3)
14.3 Subsolutions
396(5)
14.4 The IS Scheme Associated to a Subsolution
401(4)
14.5 Generalizations
405(7)
14.5.1 Functionals Besides Probabilities
405(1)
14.5.2 Continuous Time
406(2)
14.5.3 Level Crossing
408(1)
14.5.4 Path Dependent Events
409(2)
14.5.5 Markov Modulated Models
411(1)
14.6 Notes
412(1)
15 Performance of an IS Scheme Based on a Subsolution
413(26)
15.1 Statement of Resulting Performance
413(5)
15.2 Performance Bounds for the Finite-Time Problem
418(11)
15.3 Performance Bounds for the Exit Probability Problem
429(8)
15.4 Notes
437(2)
16 Multilevel Splitting
439(32)
16.1 Notation and Terminology
441(4)
16.2 Formulation of the Algorithm
445(6)
16.3 Performance Measures
451(6)
16.4 Design and Asymptotic Analysis of Splitting Schemes
457(9)
16.5 Splitting for Finite-Time Problems
466(3)
16.5.1 Subsolutions for Analysis of Metastability
467(2)
16.6 Notes
469(2)
17 Examples of Subsolutions and Their Application
471(38)
17.1 Estimating an Expected Value
472(5)
17.1.1 Problem Statement
472(1)
17.1.2 Associated PDE
472(1)
17.1.3 Component Functions
473(1)
17.1.4 Subsolutions
473(2)
17.1.5 Example
475(2)
17.2 Hitting Probabilities and Level Crossing
477(6)
17.2.1 Problem Statement
477(1)
17.2.2 Associated PDE
477(1)
17.2.3 Component Functions
477(2)
17.2.4 Subsolutions
479(1)
17.2.5 Examples
479(4)
17.3 Path-Dependent Functional
483(4)
17.3.1 Problem Formulation
484(1)
17.3.2 Subsolutions
484(1)
17.3.3 Example
485(2)
17.4 Serve-the-Longest Queue
487(9)
17.4.1 Problem Formulation
487(2)
17.4.2 Associated Rate Function
489(1)
17.4.3 Adaptations Needed for the WSLQ Model
489(2)
17.4.4 Characterization of Subsolutions
491(1)
17.4.5 Component Functions
491(1)
17.4.6 Subsolutions
492(2)
17.4.7 Example
494(2)
17.5 Jump Markov Processes with Moderate Deviation Scaling
496(6)
17.5.1 Problem Formulation
497(1)
17.5.2 Associated PDE
498(1)
17.5.3 Component Functions
499(1)
17.5.4 Subsolutions
499(1)
17.5.5 Example
500(2)
17.6 Escape from the Neighborhood of a Rest Point
502(6)
17.6.1 Problem Formulation
503(1)
17.6.2 Associated PDE
503(1)
17.6.3 Subsolutions
504(1)
17.6.4 Examples
504(4)
17.7 Notes
508(1)
Appendix A Spaces of Measures 509(8)
Appendix B Stochastic Kernels 517(6)
Appendix C Further Properties of Relative Entropy 523(10)
Appendix D Martingales and Stochastic Integration 533(8)
Appendix E Analysis and Measure Theory 541(4)
Conventions and Standard Notation 545(8)
Abbreviations 553(2)
Specialized Symbols 555(4)
References 559(12)
Index 571
Amarjit Budhiraja is a Professor of Statistics and Operations Research at the University of North Carolina at Chapel Hill. He is a Fellow of the IMS. His research interests include stochastic analysis, the theory of large deviations, stochastic networks and stochastic nonlinear filtering. Paul Dupuis is the IBM Professor of Applied Mathematics at Brown University and a Fellow of the AMS, SIAM and IMS.  His research interests include stochastic control, the theory of large deviations and numerical methods.