Course description

The Malliavin calculus extends the classical calculus of variations from deterministic functions to stochastic processes. It was introduced by Paul Malliavin in the 70’s to provide a probabilistic proof of Hormander hypoellipticity theorem. The main application of Malliavin calculus is to establish the regularity of the probability distributions of functionals of a Gaussian process. Basic examples are diffusion processes and solutions to stochastic partial differential equations. In addition to this main application and starting from the pioneering work by Ivan Nourdin and Giovanni Peccati, the Malliavin calculus, combined with Stein’s method for normal approximations, has proved to be a useful tool to derive quantitative central limit theorems.

The goal of this course is to introduce the basic elements of Malliavin calculus and discuss its applications to regularity of probability laws, normal approximations and mathematical finance.

Motivation

Malliavin calculus is an active research area in stochastic analysis, with a wide scope of applications in a number of fields including statistics, functional analysis and finance. The combination of Malliavin calculus and Stein’s method has proved to be a powerful theory leading to new results in areas as diverse as cosmology and statistical inference. In mathematical finance, the Malliavin calculus has played an important role in designing numerical computations of price sensitivities and finding hedging portfolios.

Course description

The Malliavin calculus extends the classical calculus of variations from deterministic functions to stochastic processes. It was introduced by Paul Malliavin in the 70’s to provide a probabilistic proof of Hormander hypoellipticity theorem. The main application of Malliavin calculus is to establish the regularity of the probability distributions of functionals of a Gaussian process. Basic examples are diffusion processes and solutions to stochastic partial differential equations. In addition to this main application and starting from the pioneering work by Ivan Nourdin and Giovanni Peccati, the Malliavin calculus, combined with Stein’s method for normal approximations, has proved to be a useful tool to derive quantitative central limit theorems.

The goal of this course is to introduce the basic elements of Malliavin calculus and discuss its applications to regularity of probability laws, normal approximations and mathematical finance.

Motivation

Malliavin calculus is an active research area in stochastic analysis, with a wide scope of applications in a number of fields including statistics, functional analysis and finance. The combination of Malliavin calculus and Stein’s method has proved to be a powerful theory leading to new results in areas as diverse as cosmology and statistical inference. In mathematical finance, the Malliavin calculus has played an important role in designing numerical computations of price sensitivities and finding hedging portfolios.