M 385C (57605) Theory of Probability

Time: MWF 2:00 – 3:00 pm

Room: RLM 9.166

Instructor: Gordan Zitkovic

Course Description: 

M 385C/CSE 384K "Theory of Probability I" is the first part of a two-semester introductory graduate course in probability theory. Its aim is to develop a modern and mathematically rigorous theory of probability, and at the same time to expose the student to illustrating examples and applications.

Topics covered by the course:

1.     Theoretical topics:

a.     Foundations of measure-theoretic probability: probability spaces, sigma-algebras and information, introduction to measure-theory, random variables, expectation, independence

b.     Limit theorems: modes of convergence (weak convergence of probability measures, characteristic functions), law(s) of large numbers, central limit theorems

c.     Discrete-time martingales: conditional expectation, filtrations, martingales, convergence theorems (and applications to the strong law of large numbers)

2.     Applications: a choice of topics among

a.     Gambling, insurance, finance

b.     Queuing and stochastic networks

c.     Optimal control of stochastic systems

Intended audience:

·       Graduate and advanced undergraduate students in mathematics with an interest in probability, stochastic processes, financial mathematics and applied mathematics in general, as well as

·       Graduate students in natural sciences, engineering, economics, business, or any other field where probability is used extensively

Prerequisites: Knowledge of multi-variable calculus, basic notions of real analysis (as in M365C) and linear algebra are assumed. All the measure-theory needed will be developed, so measure-theory is not a prerequisite.

 Textbook(s):

·       Richard Durrett: "Probability: Theory and Examples", Duxbury Press, Belmont, CA, second or third edition, 1996.

Note: meets with CSE 384K (68385)