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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
Folie 1
Preface
Dependence chart
Robert Brown’s new thing
Brownian motion as a Gaussian process
Constructions of Brownian motion
The canonical model
Brownian motion as a martingale
Brownian motion as a Markov process
Brownian motion and transition semigroups
The PDE connection
The variation of Brownian paths
Regularity of Brownian paths
Brownian motion as a random fractal
The growth of Brownian paths
Strassen’s functional law of the iterated logarithm
Skorokhod representation
Stochastic integrals: L2-Theory
Stochastic integrals: localization
Stochastic integrals: martingale drivers
Itô’s formula
Applications of Itô’s formula
Wiener Chaos and iterated Wiener–Itô integrals
Stochastic differential equations
Stratonovich’s stochastic calculus
On diffusions
Simulation of Brownian motion by Björn Böttcher
Appendix
Bibliography
Index