MATH 464 - Advanced Stochastic Processes

MATH 464 provides a rigorous coverage of important aspects of the theory of stochastic processes, based on measure-theory. The first part of the course addresses the most important notions related to stochastic processes in discrete-time. It mostly focuses on the random walk as a motivating example to introduce stopping times, Markov processes and martingales, together with the important results related. This prepares the audience for the second part which addresses the continuous-time stochastic processes. Mainly, the processes covered are the Poisson process and Brownian motion. In the last part of the course, different topics can be addressed among which: Brownian bridge, Gaussian processes, Point processes, stochastic calculus, stochastic differential equations, stochastic control, etc.

Prerequisites

The course doesn't have formal prerequisites. Having had a measure-theoretic probability course (MATH 463) is strongly advised.

Why is it interesting for Probabilistic Modelers?

Stochastic Processes (time-dependent) and Random Fields (space-time dependent) are at the heart of Probabilistic Modeling. This course provides a rigourous and thorough coverage of the fundamental processes encoutered in fields of applications, such as Physics, Engineering, Biology, Finance, etc. A solid knowledge of the properties of these processes is crucial for a student or researcher to be at ease when it comes to apply as appropriate model to a given situation or when it comes to developing new models. Probabilistic modelers have often learned about stochastic processes from an applied point of view.

This course aims at providing a systematic treatment to the audience, which will become familiar with the important tools and techniques required in the study of stochastic processes. The concepts will be illustrated with examples from the theory as well as from applications.

Schedule

Offered semi-annually in the Fall semester (usually on even years). Schedule varies.