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Analysis and Approximation of Rare Events

Representations and Weak Convergence Methods

  • Amarjit Budhiraja
  • Paul Dupuis
Book

Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 94)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Laplace Principle, Relative Entropy, and Elementary Examples

    1. Front Matter
      Pages 1-1
    2. Amarjit Budhiraja, Paul Dupuis
      Pages 3-29
    3. Amarjit Budhiraja, Paul Dupuis
      Pages 31-47
    4. Amarjit Budhiraja, Paul Dupuis
      Pages 49-76
  3. Discrete Time Processes

    1. Front Matter
      Pages 77-78
    2. Amarjit Budhiraja, Paul Dupuis
      Pages 79-117
    3. Amarjit Budhiraja, Paul Dupuis
      Pages 119-149
    4. Amarjit Budhiraja, Paul Dupuis
      Pages 151-179
    5. Amarjit Budhiraja, Paul Dupuis
      Pages 181-207
  4. Continuous Time Processes

    1. Front Matter
      Pages 209-210
    2. Amarjit Budhiraja, Paul Dupuis
      Pages 211-244
    3. Amarjit Budhiraja, Paul Dupuis
      Pages 261-294
    4. Amarjit Budhiraja, Paul Dupuis
      Pages 295-318
    5. Amarjit Budhiraja, Paul Dupuis
      Pages 319-342
    6. Amarjit Budhiraja, Paul Dupuis
      Pages 343-380
  5. Accelerated Monte Carlo for Rare Events

    1. Front Matter
      Pages 381-382
    2. Amarjit Budhiraja, Paul Dupuis
      Pages 383-412
    3. Amarjit Budhiraja, Paul Dupuis
      Pages 413-437
    4. Amarjit Budhiraja, Paul Dupuis
      Pages 439-469
    5. Amarjit Budhiraja, Paul Dupuis
      Pages 471-508
  6. Back Matter
    Pages 509-574

About this book

Introduction

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.

Keywords

Discrete time processes Monte Carlo Approximation large deviation principle weak convergence methods Rare events large deviation moderate deviation weak convergence relative entropy stochastic analysis representation formulas

Authors and affiliations

  • Amarjit Budhiraja
    • 1
  • Paul Dupuis
    • 2
  1. 1.Department of Statistics and Operations ResearchUniversity of North CarolinaChapel HillUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4939-9579-0
  • Copyright Information Springer Science+Business Media, LLC, part of Springer Nature 2019
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4939-9577-6
  • Online ISBN 978-1-4939-9579-0
  • Series Print ISSN 2199-3130
  • Series Online ISSN 2199-3149
  • Buy this book on publisher's site