Malliavin calculus for Lévy processes and infinite-dimensional Brownian motion : an introduction / Horst Osswald.

Format:

Book

Author/Creator:

Osswald, Horst, author.

Series:

Cambridge tracts in mathematics ; 191.

Cambridge tracts in mathematics ; 191

Language:

English

Subjects (All):

Malliavin calculus.

Lévy processes.

Brownian motion processes.

Physical Description:

1 online resource (xix, 407 pages) : digital, PDF file(s).

Other Title:

Malliavin Calculus for Lévy Processes & Infinite-Dimensional Brownian Motion

Place of Publication:

Cambridge : Cambridge University Press, 2012.

Language Note:

English

Summary:

Assuming only basic knowledge of probability theory and functional analysis, this book provides a self-contained introduction to Malliavin calculus and infinite-dimensional Brownian motion. In an effort to demystify a subject thought to be difficult, it exploits the framework of nonstandard analysis, which allows infinite-dimensional problems to be treated as finite-dimensional. The result is an intuitive, indeed enjoyable, development of both Malliavin calculus and nonstandard analysis. The main aspects of stochastic analysis and Malliavin calculus are incorporated into this simplifying framework. Topics covered include Brownian motion, Ornstein-Uhlenbeck processes both with values in abstract Wiener spaces, Lévy processes, multiple stochastic integrals, chaos decomposition, Malliavin derivative, Clark-Ocone formula, Skorohod integral processes and Girsanov transformations. The careful exposition, which is neither too abstract nor too theoretical, makes this book accessible to graduate students, as well as to researchers interested in the techniques.

Contents:

Martingales

Fourier and Laplace transformations

Abstract Wiener-Fréchet spaces

Two concepts of no-anticipation in time

Malliavin calculus on the space of real sequences

Introduction to poly-saturated models of mathematics

Extension of the real numbers

Topology

Measure and integration on Loeb spaces

From finite- to infinite-dimensional Brownian motion

The Itô integral for infinite-dimensional Brownian motion

Multiple integrals

Infinite-dimensional Ornstein-Uhlenbeck processes

Lindstrøm's construction of standard Lévy processes from discrete ones

Stochastic integration for Lévy processes

Chaos decomposition (for infinite-dimensional Brownian motion)

The Malliavin derivative

The Skorohod integral

The interplay between derivative and integral

Skorohod integral processes

Girsanov transformations

Malliavin calculus for Lévy processes (Chaos, Malliavin derivative, Clark-Ocone formula, Skorohod integral processes, Smooth representations, a communication rule for derivative and limit, product-, chainrule, Girsanov transformations)

Poly-saturated models

The existence of poly-saturated models.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-107-23038-1

1-280-48571-X

1-139-23307-6

9786613580696

1-139-23085-9

1-139-22939-7

1-139-06011-2

1-139-23230-4

1-139-23384-X

OCLC:

780425845