Brownian Motion : An Introduction to Stochastic Processes

ISBN-13: 9783110278897 / Angielski / Twarda / 2012 / 380 str.

Rene Schilling; Lothar Partzsch

Opis

Stochastic processes occur in a large number of fields in sciences and engineering, so they need to be understood by applied mathematicians, engineers and scientists alike. This work is ideal for a first course introducing the reader gently to the subject matter of stochastic processes. It uses Brownian motion since this is a stochastic process which is central to many applications and which allows for a treatment without too many technicalities. All chapters are modular and are written in a style where the lecturer can "pick and mix" topics. A "dependence chart" will guide the reader when arrange her/his own digest of material.

Spis treści

1 Robert Brown’s New Thing
2 Constructions of Brownian Motion
3 Brownian Motion in Rd
4 The Canonical Model
5 The Variation of Brownian Paths
6 Regularity of Brownian Paths
7 Brownian Motion as a Martingale
8 Brownian Motion as a Markov process

A Semigroups, Generators and Dirichlet forms
B Brownian motion and Boundary value problems

9 Stochastic Integrals: L2-Theory
10 Stochastic Integrals: beyond L2
11 Itô’s formula
12 Applications of Itô’s formula

C Elementary Theory of Stochastic Differential equations
D Introduction to Brownian local times
E Numerical Simulation of Brownian paths and Monte-Carlo methods

Appendix
1 Kolmogorov’s Existence Theorem
2 From Discrete to Continuous-Time Martingales
3 Stopping and Sampling