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.
Horst Osswald is a Professor of Mathematics at Universitat Munchen.
Part I. The Fundamental Principles: 1. Preface; 2. Martingales; 3. Fourier and Laplace transformations; 4. Abstract Wiener-Frechet 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 Ito integral for infinite-dimensional Brownian motion; 13. The iterated integral; 14. Infinite-dimensional Ornstein-Uhlenbeck processes; 15. Lindstrom's construction of standard Levy processes from discrete ones; 16. Stochastic integration for Levy 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 Levy processes; Appendix A. Poly-saturated models; Appendix B. The existence of poly-saturated models; References; Index.