Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion

After functional, measure and stochastic analysis prerequisites, the author covers chaos decomposition, Skorohod integral processes, Malliavin derivative and Girsanov transformations.

Horst Osswald (Author)

9781107016149, Cambridge University Press

Hardback, published 1 March 2012

428 pages
23.4 x 15.8 x 2.7 cm, 0.76 kg

'In addition to being self-contained, this book remains at an accessible level despite the amount of material to be assimilated on the *-extension of real numbers. It covers the main aspects of the Malliavin calculus and succeeds in providing a good global understanding …' Zentralvlatt MATH

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.

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.

Subject Areas: Probability & statistics [PBT], Functional analysis & transforms [PBKF], Calculus [PBKA], Mathematical logic [PBCD]