3 Review(s)

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 Malliav in calculus and nonstandard analysis. The main aspects of stochastic analysis and Malliav in 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. Table Of Contents 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 It 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 Malliav in 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.