Brownian Motion: An Introduction to Stochastic Processes

✍ Scribed by René L. Schilling; Lothar Partzsch; Björn Böttcher

• Publisher: De Gruyter
• Year: 2014
• Tongue: English
• Leaves: 424
• Edition: 2nd revised and extended edition
• Category: Library

✦ Synopsis

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.

TEXT TEXT

✦ Table of Contents

Preface to the second edition
Preface
Contents
Dependence chart
Index of notation
1. Robert Brown’s new thing
2. Brownian motion as a Gaussian process
2.1 The finite dimensional distributions
2.2 Brownian motion in Rd
2.3 Invariance properties of Brownian motion
3. Constructions of Brownian motion
3.1 A random orthogonal series
3.2 The Lévy–Ciesielski construction
3.3 Wiener’s construction
3.4 Lévy’s original argument
3.5 Donsker’s construction
3.6 The Bachelier–Kolmogorov point of view
4. The canonical model
4.1 Wiener measure
4.2 Kolmogorov’s construction
5. Brownian motion as a martingale
5.1 Some ‘Brownian’ martingales
5.2 Stopping and sampling
5.3 The exponential Wald identity
6. Brownian motion as a Markov process
6.1 The Markov property
6.2 The strong Markov property
6.3 Desiré André’s reflection principle
6.4 Transience and recurrence
6.5 Lévy’s triple law
6.6 An arc-sine law
6.7 Some measurability issues
7. Brownian motion and transition semigroups
7.1 The semigroup
7.2 The generator
7.3 The resolvent
7.4 The Hille-Yosida theorem and positivity
7.5 The potential operator
7.6 Dynkin’s characteristic operator
8. The PDE connection
8.1 The heat equation
8.2 The inhomogeneous initial value problem
8.3 The Feynman–Kac formula
8.4 The Dirichlet problem
9. The variation of Brownian paths
9.1 The quadratic variation
9.2 Almost sure convergence of the variation sums
9.3 Almost sure divergence of the variation sums
9.4 Lévy’s characterization of Brownian motion
10. Regularity of Brownian paths
10.1 Hölder continuity
10.2 Non-differentiability
10.3 Lévy’s modulus of continuity
11. Brownian motion as a random fractal
11.1 Hausdorff measure and dimension
11.2 The Hausdorff dimension of Brownian paths
11.3 Local maxima of a Brownian motion
11.4 On the level sets of a Brownian motion
11.5 Roots and records
12. The growth of Brownian paths
12.1 Khintchine’s law of the iterated logarithm
12.2 Chung’s ‘other’ law of the iterated logarithm
13. Strassen’s functional law of the iterated logarithm
13.1 The Cameron–Martin formula
13.2 Large deviations (Schilder’s theorem)
13.3 The proof of Strassen’s theorem
14. Skorokhod representation
15. Stochastic integrals: L2-Theory
15.1 Discrete stochastic integrals
15.2 Simple integrands
15.3 Extension of the stochastic integral to L2T
15.4 Evaluating Itô integrals
15.5 What is the closure of ST?
15.6 The stochastic integral for martingales
16. Stochastic integrals: beyond L2T
17. Itô’s formula
17.1 Itô processes and stochastic differentials
17.2 The heuristics behind Itô’s formula
17.3 Proof of Itô’s formula (Theorem 17.1)
17.4 Itô’s formula for stochastic differentials
17.5 Itô’s formula for Brownian motion in Rd
17.6 The time-dependent Itô formula
17.7 Tanaka’s formula and local time
18. Applications of Itô’s formula
18.1 Doléans–Dade exponentials
18.2 Lévy’s characterization of Brownian motion
18.3 Girsanov’s theorem
18.4 Martingale representation–1
18.5 Martingale representation – 2
18.6 Martingales as time-changed Brownian motion
18.7 Burkholder–Davis–Gundy inequalities
19. Stochastic differential equations
19.1 The heuristics of SDEs
19.2 Some examples
19.3 The general linear SDE
19.4 Transforming an SDE into a linear SDE
19.5 Existence and uniqueness of solutions
19.6 Further examples and counterexamples
19.7 Solutions as Markov processes
19.8 Localization procedures
19.9 Dependence on the initial values
20. Stratonovich’s stochastic calculus
20.1 The Stratonovich integral
20.2 Solving SDEs with Stratonovich’s calculus
21. On diffusions
21.1 Kolmogorov’s theory
21.2 Itô’s theory
22. Simulation of Brownian motion by Björn Böttcher
22.1 Introduction
22.2 Normal distribution
22.3 Brownian motion
22.4 Multivariate Brownian motion
22.5 Stochastic differential equations
22.6 Monte Carlo method
A. Appendix
A.1 Kolmogorov’s existence theorem
A.2 A property of conditional expectations
A.3 From discrete to continuous time martingales
A.4 Stopping and sampling
A.4.1 Stopping times
A.4.2 Optional sampling
A.5 Remarks on Feller processes
A.6 The Doob–Meyer decomposition
A.7 BV functions and Riemann–Stieltjes integrals
A.7.1 Functions of bounded variation
A.7.2 The Riemann–Stieltjes Integral
A.8 Some tools from analysis
A.8.1 Frostman’s theorem: Hausdorff measure, capacity and energy
A.8.2 Gronwall’s lemma
A.8.3 Completeness of the Haar functions
Bibliography
Index