This course is the natural continuation of the course “Probability” (laurea Magistrale). The objectives include the theoretical study of continuous-time stochastic processes and stochastic calculus. At the end of the course, the student is expected to understand the fundamentals of stochastic calculus for continuous processes and to be able to apply this theory in probability calculations.
Course Prerequisites
The courses of Probability and Functional Analysis of the Laurea Magistrale.
Teaching Methods
Lectures. (Exercises will be also discussed during the course).
Assessment Methods
The exam consists of an oral test. This test is aimed at verifying the level of understanding of the theoretical topics and exercises presented in class. The questions will be structured on variable difficulty in order to establish the degree of depth in the acquisition of these skills. The formulation of the grade will be obtained by considering the overall breadth and depth of learning, as well as the clarity of the presentation and the skills demonstrated in problem solving.
Texts
The main reference textbook is 1. Stochastic Calculus: An Introduction Through Theory and Exercises, P. Baldi, Springer (2017)
Other useful texts for the course are 1. Stochastic Differential Equations: An Introduction with Applications, B. Oksendal, Springer Science & Business Media (2013) 2. Brownian Motion, Martingales, and Stochastic Calculus, Jean-François Le Gall, Springer International Publishing Switzerland (2016) 3. Stochastic Differential Equations and Diffusion Processes, Nobuyuki Ikeda and Shinzo Watanabe, Vol. 24, Elsevier (2014) 4. Multidimensional Diffusion Processes, Daniel W. Stroock and S. R. Srinivasa Varadhan, Springer (2007) 5. Numerical Solution of SDE through Computer Experiments, Peter E. Kloeden, Eckhard Platen, and Henri Schurz, Springer Science & Business Media (2012)
Contents
a. Definition of continuous-time stochastic processes, Gaussian processes, and Brownian motion. Construction of Brownian motion. Review of discrete-time martingale theory and generalization to continuous-time martingales with continuous paths.
b. Definition of the Itô integral with respect to Brownian motion. Itô’s formula and its applications. Girsanov’s theorem.
c. Theory of stochastic differential equations (SDEs) driven by Brownian motion. Notions of existence and existence and uniqueness theorems. Lyapunov functions for SDEs. Relations between solutions of SDEs and parabolic partial differential equations (Kolmogorov equation).
