Malliavin Calculus and Its Applications
About this Title
David Nualart, The University of Kansas, Lawrence, KS
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2009: Volume 110
ISBNs: 978-0-8218-4779-4 (print); 978-1-4704-1568-6 (online)
MathSciNet review: MR2498953
MSC: Primary 60H07; Secondary 60-02, 91Gxx
The Malliavin calculus was developed to provide a probabilistic proof of Hörmander's hypoellipticity theorem. The theory has expanded to encompass other significant applications.
The main application of the Malliavin calculus is to establish the regularity of the probability distribution of functionals of an underlying Gaussian process. In this way, one can prove the existence and smoothness of the density for solutions of various stochastic differential equations. More recently, applications of the Malliavin calculus in areas such as stochastic calculus for fractional Brownian motion, central limit theorems for multiple stochastic integrals, and mathematical finance have emerged.
The first part of the book covers the basic results of the Malliavin calculus. The middle part establishes the existence and smoothness results that then lead to the proof of Hörmander's hypoellipticity theorem. The last part discusses the recent developments for Brownian motion, central limit theorems, and mathematical finance.
Graduate students and research mathematicians interested in probability, the Malliavin calculus, and stochastic partial differential equations.
Table of Contents
- Chapter 1. The derivative operator
- Chapter 2. The divergence operator
- Chapter 3. The Ornstein-Uhlenbeck semigroup
- Chapter 4. Sobolev spaces and equivalence of norms
- Chapter 5. Regularity of probability laws
- Chapter 6. Support properties. Density of the maximum
- Chapter 7. Application of Malliavin calculus to diffusion processes
- Chapter 8. The divergence operator as a stochastic integral
- Chapter 9. Central limit theorems and Malliavin calculus
- Chapter 10. Applications of Malliavin calculus in finance