Brownian Motion: An Introduction to Stochastic Processes

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

• Publisher: De Gruyter
• Year: 2012
• Tongue: English
• Leaves: 396
• 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.

• Gently introduces stochastic processes addressing a wide audience comprising mathematicians, economists, engineers and scientists
• Appropriate as a textbook for graduate courses, reading courses or for independent study
• Includes modular chapters and a "dependence chart" which will guide the readers when arranging their own digest of material
• More than 200 exercises (with solutions on the internet) help beginners to understand the material

✦ Table of Contents

Preface
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 Invariance properties of Brownian motion
2.3 Brownian Motion in ℝd
3 Constructions of Brownian motion
3.1 The Lévy-Ciesielski construction
3.2 Lévy’s original argument
3.3 Wiener’s construction
3.4 Donsker’s construction
3.5 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 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 Levy’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 The growth of Brownian paths
11.1 Khintchine’s Law of the Iterated Logarithm
11.2 Chung’s ‘other’ Law of the Iterated Logarithm
12 Strassen’s Functional Law of the Iterated Logarithm
12.1 The Cameron-Martin formula
12.2 Large deviations (Schilder’s theorem)
12.3 The proof of Strassen’s theorem
13 Skorokhod representation
14 Stochastic integrals: L2-Theory
14.1 Discrete stochastic integrals
14.2 Simple integrands
14.3 Extension of the stochastic integral to L2T
14.4 Evaluating Itô integrals
14.5 What is the closure of ℇT?
14.6 The stochastic integral for martingales
15 Stochastic integrals: beyond L2T
16 Itô’s formula
16.1 Itô processes and stochastic differentials
16.2 The heuristics behind Itô’s formula
16.3 Proof of Itô’s formula (Theorem 16.1)
16.4 Itô’s formula for stochastic differentials
16.5 Itô’s formula for Brownian motion in ℝd
16.6 Tanaka’s formula and local time
17 Applications of Itô’s formula
17.1 Doléans-Dade exponentials
17.2 Lévy’s characterization of Brownian motion
17.3 Girsanov’s theorem
17.4 Martingale representation – 1
17.5 Martingale representation – 2
17.6 Martingales as time-changed Brownian motion
17.7 Burkholder-Davis-Gundy inequalities
18 Stochastic differential equations
18.1 The heuristics of SDEs
18.2 Some examples
18.3 Existence and uniqueness of solutions
18.4 Solutions as Markov processes
18.5 Localization procedures
18.6 Dependence on the initial values
19 On diffusions
19.1 Kolmogorov’s theory
19.2 Itô’s theory
20 Simulation of Brownian motion
20.1 Introduction
20.2 Normal distribution
20.3 Brownian motion
20.4 Multivariate Brownian motion
20.5 Stochastic differential equations
20.6 Monte Carlo method
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 Gronwall’s lemma
A.8.2 Completeness of the Haar functions
A.8.3 A multinomial identity
Index