Loeb Measures in Practice: Recent Advances: EMS Lectures 1997 [Paperback]

This expanded version of the 1997 European Mathematical Society Lectures given by the author in Helsinki, begins with a self-contained introduction to nonstandard analysis (NSA) and the construction of Loeb Measures, which are rich measures discovered in 1975 by Peter Loeb, using techniques from NSA. Subsequent chapters sketch a range of recent applications of Loeb measures due to the author and his collaborators, in such diverse fields as (stochastic) fluid mechanics, stochastic calculus of variations ( Malliavin calculus) and the mathematical finance theory. The exposition is designed for a general audience, and no previous knowledge of either NSA or the various fields of applications is assumed.Loeb Measures: Introduction.- Nonstandard Analysis.- Construction of Loeb Measures.- Loeb Integration Theory.- Elementary Applications. Stochastic Fluid Mechanics: Introduction.- Solution of the Deterministic Navier-Stokes Equations.- Solution of the Stochastic Navier-Stokes Equations.- Stochastic Euler Equations.- Statistical Solutions.- Attractors for the Navier-Stokes Equations.- Measure Attractors for Stochastic Navier-Stokes Equations.- Stochastic Attractors for Navier-Stokes Equations.- Attractors for the 3-dimensional Stochastic Navier-Stokes Equations. Stochastic Calculus of Variations: Introduction.- Flat Integral Representation of Wiener Measure.- The Wiener Sphere.- Brownian Motion on the Wiener Sphere and the Infinite Dimensional Ornstein-Uhlenbeck Process.- Malliavin Calculus. Mathematical Finance Theory: Introduction.- The Cox-Ross-Rubinstein Models.- Options and Contingent Claims.- The Black-Scholes Model...
The complete table of contents can be found on the Internet: http://www.springer.deIncludes supplementary material: sn.pub/extras