Stochastic Processes and their Applications

This class is a general theoretical introduction to stochastic processes and to some of their applications. The class starts with the classic elements of the general theory of semimartingales, such as stopping times and Brownian motion. It constructs the Wiener integral and then the Ito integral. Key results such as Ito’s lemma and Girsanov’s theorem are then introduced and some illustrations are provided.

This class offers an introduction to stable distributions and stable processes, and more generally to fractals and self-similarity. It also provides a description of infinitely divisible distributions and of Lévy processes – from simple Poisson processes to the more complex and contemporaneous CGMY processes. More generally, this class develops a Fourier vision of the theory of stochastic processes.