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MATH332 THEORETICAL ASPECTS OF STOCHASTIC PROCESSES

Course Objectives

This course is intended to serve as a first introduction to stochastic processes, with an emphasis on theoretical aspects and proofs, without requiring full measure theoretical foundations, but only requiring a background course in basic probability theory such as Math 301. The course provides the students with an introduction to examples of stochastic processes much sought for in the industry, such as discrete and continuous time Markov chains, martingales and Brownian motion. In addition, connection with other areas of mathematics such as the amenability problem in group theory and various applications in random geometry will be discussed. The course will equip students intending to continue in diverse pure or applied areas such as group theory, random geometry, financial mathematics, dynamical systems, probabilistic aspects of data science or telecommunication networks, with essentials of the subject.

Course Content

Introduction to stochastic processes. Emergence and applications of stochastic processes in various areas of mathematics such as geometry and group theory. Finite and countable Markov chains. Classification of states with proofs. Continuous-time Markov chains; Poisson process. Conditional expectation. Martingales. Brownian motion. Fractal nature of zero sets of Brownian motion.

Course Learning Outcomes

At the end of the course, students are expected to:

• calculate the large time behavior of finite Markov chains and classify their states,
• decide whether a countable Markov chain is recurrent or transient,
• use the optional sampling theorem for martingales to compute expectations,
• understand the basics of Brownian motion in one and several dimensions.