In 1976 Paul Malliavin proposed in the paper "Stochastic calculus of variation and hypoelliptic operators" a probabilistic proof of Hormander's "sum of square" theorem. His proof was based on a new infinite dimensional differential calculus on the Wiener space. The theory was further developed by Stroock, Bismut and Watanabe, among others, to become what is nowadays known as the Malliavin calculus. This calculus has become a fundamental tool in the theory of stochastic (partial) differential equations and has found important applications in mathematical finance. This short course aims to provide a coincise introduction to the subject together with a sketch of Malliavin's proof of Hormander's theorem. Few remarks on the applications in mathematical finance will also be provided.

In 1976 Paul Malliavin proposed in the paper "Stochastic calculus of variation and hypoelliptic operators" a probabilistic proof of Hormander's "sum of square" theorem. His proof was based on a new infinite dimensional differential calculus on the Wiener space. The theory was further developed by Stroock, Bismut and Watanabe, among others, to become what is nowadays known as the Malliavin calculus. This calculus has become a fundamental tool in the theory of stochastic (partial) differential equations and has found important applications in mathematical finance. This short course aims to provide a coincise introduction to the subject together with a sketch of Malliavin's proof of Hormander's theorem. Few remarks on the applications in mathematical finance will also be provided