This textbook provides an introductory course on Malliavin calculus intended to prepare the interested reader for further study of existing monographs on the subject such as Bichteler et al. (1987), Malliavin (1991), Sanz-Sol'e (2005), Malliavin and Thalmaier (2005), Nualart (2006), Di Nunno et al. (2009), Nourdin and Peccati (2012), and Ishikawa (2016), among others. Moreover, it contains recent applications of Malliavin calculus, including 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 an advanced probability course that includes laws of large numbers and central limit theorems, martingales, and Markov processes.
The Malliavin calculus is an infinite-dimensional differential calculus on Wiener space, first introduced by Paul Malliavin in the 1970s with the aim of giving a probabilistic proof of H‥ormander's hypoellipticity theorem; see Malliavin (1978a, b, c). The theory was further developed, see e.g. Shigekawa (1980), Bismut (1981), Stroock (1981a, b), and Ikeda and Watanabe (1984), and since then many new applications have appeared.
Chapters 1 and 2 give an introduction to stochastic calculus with respect to Brownian motion, as developed by Itô (1944). The purpose of this calculus is to construct stochastic integrals for adapted and square integrable processes and to develop a change-of-variable formula.
Chapters 3, 4, and 5 present the main operators of the Malliavin calculus, which are the derivative, the divergence, the generator of the Ornstein– Uhlenbeck semigroup, and the corresponding Sobolev norms. In Chapter 4, multiple stochastic integrals are constructed following Itô (1951), and the orthogonal decomposition of square integrable random variables due to Wiener (1938) is derived. These concepts play a key role in the development of further properties of the Malliavin calculus operators. In particular, Chapter 5 contains an integration-by-parts formula that relates the three operators, which is crucial for applications. In particular, it allows us to prove a density formula due to Nourdin and Viens (2009).
Chapters 6, 7, and 8 are devoted to different applications of the Malliavin calculus for Brownian motion. Chapter 6 presents two different stochastic integral representations: the first is the well-known Clark–Ocone formula, and the second uses the inverse of the Ornstein–Ulhenbeck generator.