Stochastic Calculus and Applications

The objective of the course is to provide an introduction to stochastic calculus, more specifically Ito calculus. Starting from the main definitions and results of stochastic processes theory, in particular of the Brownian motion, we will introduce Ito's stochastic integral and we will investigate its main properties. Moreover, we will study the stochastic differential equations, showing existence and uniqueness of their solutions in the Lipschitz case. Finally, we will present some applications in analysis, e.g. Feynman-Kac formula, which highlight the link between stochastic differential equations and partial differential equations.

The students will learn the notion of stochastic integral, its main properties and some fundamental results of stochastic calculus based on it. Moreover, he/she will be able to solve some classes of stochastic differential equations, to study their main properties also by exploiting the link between them and partial differential equations.