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Brownian Motion, 2nd Edition

Book Description

Brownian motion is one of the most important stochastic processes in continuous time and with continuous state space. Within the realm of stochastic processes, Brownian motion is at the intersection of Gaussian processes, martingales, Markov processes, diffusions and random fractals, and it has influenced the study of these topics. Its central position within mathematics is matched by numerous applications in science, engineering and mathematical finance.

Often textbooks on probability theory cover, if at all, Brownian motion only briefly. On the other hand, there is a considerable gap to more specialized texts on Brownian motion which is not so easy to overcome for the novice. The authors’ aim was to write a book which can be used as an introduction to Brownian motion and stochastic calculus, and as a first course in continuous-time and continuous-state Markov processes. They also wanted to have a text which would be both a readily accessible mathematical back-up for contemporary applications (such as mathematical finance) and a foundation to get easy access to advanced monographs.

This textbook, tailored to the needs of graduate and advanced undergraduate students, covers Brownian motion, starting from its elementary properties, certain distributional aspects, path properties, and leading to stochastic calculus based on Brownian motion. It also includes numerical recipes for the simulation of Brownian motion.

Table of Contents

  1. Also of Interest
  2. Title Page
  3. Copyright Page
  4. Preface to the second edition
  5. Preface
  6. Table of Contents
  7. Dependence chart
  8. Index of notation
  9. 1 Robert Brown’s new thing
    1. Problems
  10. 2 Brownian motion as a Gaussian process
    1. 2.1 The finite dimensional distributions
    2. 2.2 Brownian motion in
    3. 2.3 Invariance properties of Brownian motion
    4. Problems
  11. 3 Constructions of Brownian motion
    1. 3.1 A random orthogonal series
    2. 3.2 The Lévy-Ciesielski construction
    3. 3.3 Wiener’s construction
    4. 3.4 Lévy’s original argument
    5. 3.5 Donsker’s construction
    6. 3.6 The Bachelier-Kolmogorov point of view
    7. Problems
  12. 4 The canonical model
    1. 4.1 Wiener measure
    2. 4.2 Kolmogorov’s construction
    3. Problems
  13. 5 Brownian motion as a martingale
    1. 5.1 Some ‘Brownian’ martingales
    2. 5.2 Stopping and sampling
    3. 5.3 The exponential Wald identity
    4. Problems
  14. 6 Brownian motion as a Markov process
    1. 6.1 The Markov property
    2. 6.2 The strong Markov property
    3. 6.3 Desire André’s reflection principle
    4. 6.4 Transience and recurrence
    5. 6.5 Lévy’s triple law
    6. 6.6 An arc-sine law
    7. 6.7 Some measurability issues
    8. Problems
  15. 7 Brownian motion and transition semigroups
    1. 7.1 The semigroup
    2. 7.2 The generator
    3. 7.3 The resolvent
    4. 7.4 The Hille-Yosida theorem and positivity
    5. 7.5 The potential operator
    6. 7.6 Dynkin’s characteristic operator
    7. Problems
  16. 8 The PDE connection
    1. 8.1 The heat equation
    2. 8.2 The inhomogeneous initial value problem
    3. 8.3 The Feynman-Kac formula
    4. 8.4 The Dirichlet problem
    5. Problems
  17. 9 The variation of Brownian paths
    1. 9.1 The quadratic variation
    2. 9.2 Almost sure convergence of the variation sums
    3. 9.3 Almost sure divergence of the variation sums
    4. 9.4 Lévy’s characterization of Brownian motion
    5. Problems
  18. 10 Regularity of Brownian paths
    1. 10.1 Hölder continuity
    2. 10.2 Non-differentiability
    3. 10.3 Lévy’s modulus of continuity
    4. Problems
  19. 11 Brownian motion as a random fractal
    1. 11.1 Hausdorff measure and dimension
    2. 11.2 The Hausdorff dimension of Brownian paths
    3. 11.3 Local maxima of a Brownian motion
    4. 11.4 On the level sets of a Brownian motion
    5. 11.5 Roots and records
    6. Problems
  20. 12 The growth of Brownian paths
    1. 12.1 Khintchine’s law of the iterated logarithm
    2. 12.2 Chung’s ‘other’ law of the iterated logarithm
    3. Problems
  21. 13 Strassen’s functional law of the iterated logarithm
    1. 13.1 The Cameron-Martin formula
    2. 13.2 Large deviations (Schilder’s theorem)
    3. 13.3 The proof of Strassen’s theorem
    4. Problems
  22. 14 Skorokhod representation
    1. Problems
  23. 15 Stochastic integrals: L2-Theory
    1. 15.1 Discrete stochastic integrals
    2. 15.2 Simple integrands
    3. 15.3 Extension of the stochastic integral to
    4. 15.4 Evaluating Itô integrals
    5. 15.5 What is the closure of ST?
    6. 15.6 The stochastic integral for martingales
    7. Problems
  24. 16 Stochastic integrals: beyond
    1. Problems
  25. 17 Itô’s formula
    1. 17.1 Itô processes and stochastic differentials
    2. 17.2 The heuristics behind Itô’s formula
    3. 17.3 Proof of Itô’s formula (Theorem 17.1)
    4. 17.4 Itô’s formula for stochastic differentials
    5. 17.5 Itô’s formula for Brownian motion in ℝd
    6. 17.6 The time-dependent Itô formula
    7. 17.7 Tanaka’s formula and local time
    8. Problems
  26. 18 Applications of Itô’s formula
    1. 18.1 Doléans–Dade exponentials
    2. 18.2 Lévy’s characterization of Brownian motion
    3. 18.3 Girsanov’s theorem
    4. 18.4 Martingale representation – 1
    5. 18.5 Martingale representation — 2
    6. 18.6 Martingales as time-changed Brownian motion
    7. 18.7 Burkholder-Davis-Gundy inequalities
    8. Problems
  27. 19 Stochastic differential equations
    1. 19.1 The heuristics of SDEs
    2. 19.2 Some examples
    3. 19.3 The general linear SDE
    4. 19.4 Transforming an SDE into a linear SDE
    5. 19.5 Existence and uniqueness of solutions
    6. 19.6 Further examples and counterexamples
    7. 19.7 Solutions as Markov processes
    8. 19.8 Localization procedures
    9. 19.9 Dependence on the initial values
    10. Problems
  28. 20 Stratonovich’s stochastic calculus
    1. 20.1 The Stratonovich integral
    2. 20.2 Solving SDEs with Stratonovich’s calculus
    3. Problems
  29. 21 On diffusions
    1. 21.1 Kolmogorov’s theory
    2. 21.2 Itô’s theory
    3. Problems
  30. 22 Simulation of Brownian motion
    1. 22.1 Introduction
    2. 22.2 Normal distribution
    3. 22.3 Brownian motion
    4. 22.4 Multivariate Brownian motion
    5. 22.5 Stochastic differential equations
    6. 22.6 Monte Carlo method
  31. A Appendix
    1. A.1 Kolmogorov’s existence theorem
    2. A.2 A property of conditional expectations
    3. A.3 From discrete to continuous time martingales
    4. A.4 Stopping and sampling
      1. A.4.1 Stopping times
      2. A.4.2 Optional sampling
    5. A.5 Remarks on Feller processes
    6. A.6 The Doob-Meyer decomposition
    7. A.7 BV functions and Riemann-Stieltjes integrals
      1. A.7.1 Functions of bounded variation
      2. A.7.2 The Riemann-Stieltjes Integral
    8. A.8 Some tools from analysis
      1. A.8.1 Frostman’s theorem: Hausdorff measure, capacity and energy
      2. A.8.2 Gronwall’s lemma
      3. A.8.3 Completeness of the Haar functions
  32. Index