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.
Author(s): A. Budhiraja , P. Dupuis
Series: Probability Theory and Stochastic Modelling 94
Edition: 1
Publisher: Springer
Year: 2019
Language: English
Pages: 577
Front Matter ....Pages i-xix
Front Matter ....Pages 1-1
General Theory (Amarjit Budhiraja, Paul Dupuis)....Pages 3-29
Relative Entropy and Tightness of Measures (Amarjit Budhiraja, Paul Dupuis)....Pages 31-47
Examples of Representations and Their Application (Amarjit Budhiraja, Paul Dupuis)....Pages 49-76
Front Matter ....Pages 77-78
Recursive Markov Systems with Small Noise (Amarjit Budhiraja, Paul Dupuis)....Pages 79-117
Moderate Deviations for Recursive Markov Systems (Amarjit Budhiraja, Paul Dupuis)....Pages 119-149
Empirical Measure of a Markov Chain (Amarjit Budhiraja, Paul Dupuis)....Pages 151-179
Models with Special Features (Amarjit Budhiraja, Paul Dupuis)....Pages 181-207
Front Matter ....Pages 209-210
Representations for Continuous Time Processes (Amarjit Budhiraja, Paul Dupuis)....Pages 211-244
Abstract Sufficient Conditions for Large and Moderate Deviations in the Small Noise Limit (Amarjit Budhiraja, Paul Dupuis)....Pages 245-260
Large and Moderate Deviations for Finite Dimensional Systems (Amarjit Budhiraja, Paul Dupuis)....Pages 261-294
Systems Driven by an Infinite Dimensional Brownian Noise (Amarjit Budhiraja, Paul Dupuis)....Pages 295-318
Stochastic Flows of Diffeomorphisms and Image Matching (Amarjit Budhiraja, Paul Dupuis)....Pages 319-342
Models with Special Features (Amarjit Budhiraja, Paul Dupuis)....Pages 343-380
Front Matter ....Pages 381-382
Rare Event Monte Carlo and Importance Sampling (Amarjit Budhiraja, Paul Dupuis)....Pages 383-412
Performance of an IS Scheme Based on a Subsolution (Amarjit Budhiraja, Paul Dupuis)....Pages 413-437
Multilevel Splitting (Amarjit Budhiraja, Paul Dupuis)....Pages 439-469
Examples of Subsolutions and Their Application (Amarjit Budhiraja, Paul Dupuis)....Pages 471-508
Back Matter ....Pages 509-574
