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 continuoustime and continuousstate Markov processes. They also wanted to have a text which would be both a readily accessible mathematical backup 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
 Also of Interest
 Title Page
 Copyright Page
 Preface to the second edition
 Preface
 Table of Contents
 Dependence chart
 Index of notation
 1 Robert Brown’s new thing
 2 Brownian motion as a Gaussian process
 3 Constructions of Brownian motion
 4 The canonical model
 5 Brownian motion as a martingale
 6 Brownian motion as a Markov process
 7 Brownian motion and transition semigroups
 8 The PDE connection
 9 The variation of Brownian paths
 10 Regularity of Brownian paths
 11 Brownian motion as a random fractal
 12 The growth of Brownian paths
 13 Strassen’s functional law of the iterated logarithm
 14 Skorokhod representation
 15 Stochastic integrals: L2Theory
 16 Stochastic integrals: beyond

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 timedependent Itô formula
 17.7 Tanaka’s formula and local time
 Problems
 18 Applications of Itô’s formula

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
 Problems
 20 Stratonovich’s stochastic calculus
 21 On diffusions
 22 Simulation of Brownian motion
 A Appendix
 Index
Product Information
 Title: Brownian Motion, 2nd Edition
 Author(s):
 Release date: August 2014
 Publisher(s): De Gruyter
 ISBN: 9783110373981