Horst Osswald

Malliavin Calculus for Levy Processes and Infinite-Dimensional Brownian Motion (eBook, PDF)

• Format: PDF

Produktbeschreibung

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, Levy 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.

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Produktdetails

• Verlag: Cambridge University Press
• Erscheinungstermin: 1. März 2012
• Englisch
• ISBN-13: 9781139227865
• Artikelnr.: 38444709

Autorenporträt

Horst Osswald is a Professor of Mathematics at Universität München.

Inhaltsangabe

Part I. The Fundamental Principles: 1. Preface
2. Martingales
3. Fourier and Laplace transformations
4. Abstract Wiener-Fréchet spaces
5. Two concepts of no-anticipation in time
6. Malliavin calculus on the space of real sequences
7. Introduction to poly-saturated models of mathematics
8. Extension of the real numbers and properties
9. Topology
10. Measure and integration on Loeb spaces
Part II. An Introduction to Finite- and Infinite-Dimensional Stochastic Analysis: 11. From finite- to infinite-dimensional Brownian motion
12. The Itô integral for infinite-dimensional Brownian motion
13. The iterated integral
14. Infinite-dimensional Ornstein-Uhlenbeck processes
15. Lindstrøm's construction of standard Lévy processes from discrete ones
16. Stochastic integration for Lévy processes
Part III. Malliavin Calculus: 17. Chaos decomposition
18. The Malliavin derivative
19. The Skorokhod integral
20. The interplay between derivative and integral
21. Skorokhod integral processes
22. Girsanov transformation
23. Malliavin calculus for Lévy processes
Appendix A. Poly-saturated models
Appendix B. The existence of poly-saturated models
References
Index.