Stochastic Calculus and Applications

The main scope of the course is to give an introduction to the methods of stochastic calculus, with particular attention to the Ito's calculus.
From definitions and fundamental results of the theory of the stochastic processes, with particular attention to the class of Markov processes, and the of the Wiener process, the students are guided to the formulation of the systems of stochastic differential equations of the Ito's type. A construction of the Ito's integral both as L2 limit and limit in probability is given. The martingality properties of the Itos's process, following the one of the Wiener process are analyzed. Particular interest is devoted to the analysis of the stochastic differential equations and the links to the PDEs.
As a complement of the theory, a simulation laboratory is based on the concept of the learning by doing: via simulation some of the most important properties of the stochastic processes and in particular of the Wiener process are guessed; the possible counterpart of some deterministic models, and the numerical solution of some PDE via the simulation of SDE system are discussed. Particular attention to some application in Biology, Medicine and Finance is given.

Student learn how to treat and discuss the mail properties of the Markov stochastic processes and of the Winer process, in particular. He is able to understan the main probabilistic consequences of the constructiin of the Ito stochastic integral, above all the ones related to the martigales. He gain a knowledge of the stochastic differential equation and their relation to PDEs.
Besides the theoretical knowledge, he learn how it is possible to introduce the randomness modelling some situation already known from the deterministic point of view. He knows how simulate a system of SDEs and quantify the properties of the solutions via some statistical procedure.