The Malliavin calculus is an infinite-dimensional differential calculus on Wiener space, that was first introduced by Paul Malliavin in the 70’s, with the aim of giving a probabilistic proof of Hörmander’s theorem. This theory was then further developed, and since then, many new applications of this calculus have appeared.
This textbook provides an introductory course on Malliavin calculus intended to prepare the interested reader for further study of existing monographs on the subject. Moreover, it contains recent applications of Malliavin calculus: density formulas, central limit theorems for functionals of Gaussian processes, theorems on the convergence of densities, noncentral limit theorems, and Malliavin calculus for jump processes.
Recommended prior knowledge would be and advanced probability course.
The book is organized as follows. Chapters 1 and 2, give an introduction to stochastic calculus with respect to Brownian motion. Chapters 3, 4, and 5 present the main operators of the Malliavin calculus. Chapters 6, 7 and 8 are devoted to different applications of the Malliavin calculus for Brownian motion. Chapter 9, 10 and 11 develop Malliavin calculus for Poisston random measures.