The course on Stochastic Processes is mainly focused on an introductory part finalized to recover essentials of measure theory dedicated to the probability measure, and on a wide study of the Brownian motion due its key role in theory of stochastic processes. Indeed, many details will be provided about martingale and Markov properties of such a process and of processes obtained by transformations from it. Stopping times, filtrations, conditional expectations are a consistent part of the course. The course also includes the construction of the stochastic integral respect to the Brownian motion and essential hints on stochastic differential equations. Many examples and exercises are also given to clarify the mathematical aspects and potential applications of such contents in the modeling context. The whole theoretical apparatus follows the classical Kolmogorov setting. An illuminating and referring guide for the contents of such a course is the book of Paolo Baldi, “Stochastic Calculus, An Introduction Through Theory and Exercises”, Springer, 2017, that can also be considered for further insights.
