Brownian Motion: A Guide to Random Processes and Stochastic Calculus

✍ Scribed by René L. Schilling; Björn Böttcher

• Publisher: De Gruyter
• Year: 2021
• Tongue: English
• Leaves: 534
• Edition: 3rd Edition
• Category: Library

✦ Synopsis

Stochastic processes occur everywhere in the sciences, economics and engineering, and they need to be understood by (applied) mathematicians, engineers and scientists alike. This book gives a gentle introduction to Brownian motion and stochastic processes, in general. Brownian motion plays a special role, since it shaped the whole subject, displays most random phenomena while being still easy to treat, and is used in many real-life models. Im this new edition, much material is added, and there are new chapters on ''Wiener Chaos and Iterated Itô Integrals'' and ''Brownian Local Times''.

3rd revised and extended edition

More than 200 exercises

Solutions included

For mathematicians, economists, engineers and scientists

✦ Table of Contents

Preface
Contents
Dependence chart
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: L²-Theory
16 Stochastic integrals: localization
17 Stochastic integrals: martingale drivers
18 Itô’s formula
19 Applications of Itô’s formula
20 Wiener Chaos and iterated Wiener–Itô integrals
21 Stochastic differential equations
22 Stratonovich’s stochastic calculus
23 On diffusions
24 Simulation of Brownian motion by Björn Böttcher
A Appendix
Bibliography
Index