Brownian Motion - A Guide to Random Processes and Stochastic Calculus

Description

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 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