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Brownian Models of Performance and Control

Book Description

Direct and to the point, this book from one of the field's leaders covers Brownian motion and stochastic calculus at the graduate level, and illustrates the use of that theory in various application domains, emphasizing business and economics. The mathematical development is narrowly focused and briskly paced, with many concrete calculations and a minimum of abstract notation. The applications discussed include: the role of reflected Brownian motion as a storage model, queueing model, or inventory model; optimal stopping problems for Brownian motion, including the influential McDonald-Siegel investment model; optimal control of Brownian motion via barrier policies, including optimal control of Brownian storage systems; and Brownian models of dynamic inference, also called Brownian learning models, or Brownian filtering models.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. Guide to Notation and Terminology
  8. 1. Brownian Motion
    1. 1.1 Wiener’s theorem
    2. 1.2 Quadratic variation and local time
    3. 1.3 Strong Markov property
    4. 1.4 Brownian martingales
    5. 1.5 Two characterizations of Brownian motion
    6. 1.6 The innovation theorem
    7. 1.7 A joint distribution (Reflection principle)
    8. 1.8 Change of drift as change of measure
    9. 1.9 A hitting time distribution
    10. 1.10 Reflected Brownian motion
    11. 1.11 Problems and complements
  9. 2. Stochastic Storage Models
    1. 2.1 Buffered stochastic flow
    2. 2.2 The one-sided reflection mapping
    3. 2.3 Finite buffer capacity
    4. 2.4 The two-sided reflection mapping
    5. 2.5 Measuring system performance
    6. 2.6 Brownian storage models
    7. 2.7 Problems and complements
  10. 3. Further Analysis of Brownian Motion
    1. 3.1 Introduction
    2. 3.2 The backward and forward equations
    3. 3.3 Hitting time problems
    4. 3.4 Expected discounted costs
    5. 3.5 One absorbing barrier
    6. 3.6 Two absorbing barriers
    7. 3.7 More on reflected Brownian motion
    8. 3.8 Problems and complements
  11. 4. Stochastic Calculus
    1. 4.1 Introduction
    2. 4.2 First definition of the stochastic integral
    3. 4.3 An illuminating example
    4. 4.4 Final definition of the integral
    5. 4.5 Stochastic differential equations
    6. 4.6 Simplest version of Itô’s formula
    7. 4.7 The multi-dimensional Itô formula
    8. 4.8 Tanaka’s formula and local time
    9. 4.9 Another generalization of Itô’s formula
    10. 4.10 Integration by parts (Special cases)
    11. 4.11 Differential equations for Brownian motion
    12. 4.12 Problems and complements
  12. 5. Optimal Stopping of Brownian Motion
    1. 5.1 A general problem
    2. 5.2 Continuation costs
    3. 5.3 McDonald–Siegel investment model
    4. 5.4 An investment problem with costly waiting
    5. 5.5 Some general theory
    6. 5.6 Sources and literature
    7. 5.7 Problems and complements
  13. 6. Reflected Brownian Motion
    1. 6.1 Strong Markov property
    2. 6.2 Application of Itô’s formula
    3. 6.3 Expected discounted costs
    4. 6.4 Regenerative structure
    5. 6.5 The steady-state distribution
    6. 6.6 The case of a single barrier
    7. 6.7 Problems and complements
  14. 7. Optimal Control of Brownian Motion
    1. 7.1 Impulse control with discounting
    2. 7.2 Control band policies
    3. 7.3 Optimal policy parameters
    4. 7.4 Impulse control with average cost criterion
    5. 7.5 Relative cost functions
    6. 7.6 Average cost optimality
    7. 7.7 Instantaneous control with discounting
    8. 7.8 Instantaneous control with average cost criterion
    9. 7.9 Cash management
    10. 7.10 Sources and literature
    11. 7.11 Problems and complements
  15. 8. Brownian Models of Dynamic Inference
    1. 8.1 Drift-rate uncertainty in a Bayesian framework
    2. 8.2 Binary prior distribution
    3. 8.3 Brownian sequential detection
    4. 8.4 General finite prior distribution
    5. 8.5 Gaussian prior distribution
    6. 8.6 A partially observed Markov chain
    7. 8.7 Change-point detection
  16. 9. Further Examples
    1. 9.1 Ornstein–Uhlenbeck process
    2. 9.2 Probability of ruin with compounding assets
    3. 9.3 RBM with killing at the boundary
    4. 9.4 Brownian motion with an interval removed
    5. 9.5 Brownian motion with sticky reflection
    6. 9.6 A capacity expansion model
    7. 9.7 Problems and complements
  17. Appendix A. Stochastic Processes
    1. A.1 A filtered probability space
    2. A.2 Random variables and stochastic processes
    3. A.3 A canonical example
    4. A.4 Two martingale theorems
    5. A.5 A version of Fubini’s theorem
  18. Appendix B. Real Analysis
    1. B.1 Absolutely continuous functions
    2. B.2 VF functions
    3. B.3 Riemann–Stieltjes integration
    4. B.4 The Riemann–Stieltjes chain rule
    5. B.5 Notational conventions for integrals
  19. References
  20. Index