Analysis and Approximation of Rare Events: Representations and Weak Convergence Methods
by Amarjit Budhiraja (Author), Paul Dupuis (Author)

About the Author
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.

About this book
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.

Brief contents
Part I Laplace Principle, Relative Entropy, and Elementary Examples
1 General Theory 3
　　　　1.1 Large Deviation Principle 3
　　　　1.2 An Equivalent Formulation of the Large Deviation Principle 7
　　　　1.3 Basic Results in the Theory 20
　　　　1.4 Notes 29
2 Relative Entropy and Tightness of Measures 31
　　　　2.1 Properties of Relative Entropy 31
　　　　2.2 Tightness of Probability Measures 44
　　　　2.3 Notes 47
3 Examples of Representations and Their Application 49
　　　　3.1 Representation for an IID Sequence 49
　　　　3.2 Representation for Functionals of Brownian Motion 60
　　　　3.3 Representation for Functionals of a Poisson Process 69
　　　　3.4 Notes 75
Part II Discrete Time Processes
4 Recursive Markov Systems with Small Noise 79
　　　　4.1 Process Model 79
　　　　4.2 The Representation 81
　　　　4.3 Form of the Rate Function 83
　　　　4.4 Statement of the LDP 84
　　　　4.5 Laplace Upper Bound 86
　　　　4.6 Properties of Lex; bT 92
　　　　4.7 Laplace Lower Bound Under Condition 4.7 98
　　　　4.8 Laplace Lower Bound Under Condition 4.8 102
　　　　4.9 Notes 117
5 Moderate Deviations for Recursive Markov Systems 119
　　　　5.1 Assumptions, Notation, and Theorem Statement 121
　　　　5.2 The Representation 124
　　　　5.3 Tightness and Limits for Controlled Processes 125
　　　　5.4 Laplace Upper Bound 141
　　　　5.5 Laplace Lower Bound 142
　　　　5.6 Notes 149
6 Empirical Measure of a Markov Chain 151
　　　　6.1 Applications 151
　　　　6.2 The Representation 153
　　　　6.3 Form of the Rate Function 154
　　　　6.4 Assumptions and Statement of the LDP 156
　　　　6.5 Properties of the Rate Function 158
　　　　6.6 Tightness and Weak Convergence 160
　　　　6.7 Laplace Upper Bound 162
　　　　6.8 Laplace Lower Bound 163
　　　　6.9 Uniform Laplace Principle 173
　　　　6.10 Noncompact State Space 174
　　　　6.11 Notes 178
7 Models with Special Features 181
　　　　7.1 Introduction 181
　　　　7.2 Occupancy Models 182
　　　　7.3 Two Scale Recursive Markov Systems with Small Noise 203
　　　　7.4 Notes 206
Part III Continuous Time Processes
8 Representations for Continuous Time Processes 211
　　　　8.1 Representation for Infinite Dimensional Brownian Motion. 212
　　　　8.2 Representation for Poisson Random Measure 225
　　　　8.3 Representation for Functionals of PRM and Brownian　Motion 242
　　　　8.4 Notes 243
9 Abstract Sufficient Conditions for Large and Moderate　Deviations in the Small Noise Limit 245
　　　　9.1 Definitions and Notation 246
　　　　9.2 Abstract Sufficient Conditions for LDP and MDP 247
　　　　9.3 Proof of the Large Deviation Principle 253
　　　　9.4 Proof of the Moderate Deviation Principle 255
　　　　9.5 Notes 259
10 Large and Moderate Deviations for Finite Dimensional　Systems 261
　　　　10.1 Small Noise Jump-Diffusion 262
　　　　10.2 An LDP for Small Noise Jump-Diffusions 263
　　　　10.3 An MDP for Small Noise Jump-Diffusions 278
　　　　10.4 Notes 293
11 Systems Driven by an Infinite Dimensional Brownian Noise 295
　　　　11.1 Formulations of Infinite Dimensional Brownian Motion 296
　　　　11.2 General Sufficient Condition for an LDP 302
　　　　11.3 Reaction–Diffusion SPDE 306
　　　　11.4 Notes 317
12 Stochastic Flows of Diffeomorphisms and Image Matching 319
　　　　12.1 Notation and Definitions 321
　　　　12.2 Statement of the LDP 324
　　　　12.3 Weak Convergence for Controlled Flows 328
　　　　12.4 Application to Image Analysis 336
　　　　12.5 Notes 342
13 Models with Special Features 343
　　　　13.1 Introduction 343
　　　　13.2 A Model with Discontinuous Statistics-Weighted　Serve-the-Longest Queue. 344
　　　　13.3 A Class of Pure Jump Markov Processes 365
　　　　13.4 Notes 380
Part IV Accelerated Monte Carlo for Rare Events
14 Rare Event Monte Carlo and Importance Sampling 383
　　　　14.1 Example of a Quantity to be Estimated 383
　　　　14.2 Importance Sampling 387
　　　　14.3 Subsolutions 396
　　　　14.4 The IS Scheme Associated to a Subsolution 401
　　　　14.5 Generalizations 405
　　　　14.6 Notes 412
15 Performance of an IS Scheme Based on a Subsolution 413
　　　　15.1 Statement of Resulting Performance 413
　　　　15.2 Performance Bounds for the Finite-Time Problem 418
　　　　15.3 Performance Bounds for the Exit Probability Problem. 429
　　　　15.4 Notes 437
16 Multilevel Splitting 439
　　　　16.1 Notation and Terminology 441
　　　　16.2 Formulation of the Algorithm 445
　　　　16.3 Performance Measures 451
　　　　16.4 Design and Asymptotic Analysis of Splitting Schemes 457
　　　　16.5 Splitting for Finite-Time Problems 466
　　　　16.6 Notes 469
17 Examples of Subsolutions and Their Application 471
　　　　17.1 Estimating an Expected Value 472
　　　　17.2 Hitting Probabilities and Level Crossing. 477
　　　　17.3 Path-Dependent Functional 483
　　　　17.4 Serve-the-Longest Queue. 487
　　　　17.5 Jump Markov Processes with Moderate Deviation Scaling 496
　　　　17.6 Escape from the Neighborhood of a Rest Point 502
　　　　17.7 Notes 508
Appendix A: Spaces of Measures. 509
Appendix B: Stochastic Kernels. 517
Appendix C: Further Properties of Relative Entropy 523
Appendix D: Martingales and Stochastic Integration. 533

Series: Probability Theory and Stochastic Modelling (Book 94)
Pages: 574 pages
Publisher: Springer; 1st ed. 2019 edition (August 11, 2019)
Language: English
ISBN-10: 1493995774
ISBN-13: 978-1493995776