Math 562: Theory of Probability II (Fall 2018)

Goals and topics

This is the second half of the basic graduate course in probability theory. The goal of this course is to understand the basic theory of stochastic calculus. We will cover the following topics:

(1) Brownian motion;
(2) Continuous Time Matingales;
(3) Markov processes;
(4) Stochastic Integrals;
(5) Ito's formula;
(6) Representation of Martingales;
(7) Girsanov theorem and
(8) Stochastic Differential Equations.

If time allows, we will give a brief introduction to mathematical finance.

Logistics

Instructor Partha Dey
Office341A Illini Hall
ContactBy email with subject line: "Math 562:"
Class TR 9:30am -10:50am in 441 Altgeld Hall.
Office Hrs By appointment: for instance, after class (11:00am - 11:50am) is a good time to catch me between my office and the classroom. I will be happy to answer your questions in my office anytime as long as I'm not otherwise engaged.
References 1. J.F. Le Gall: Brownian motion, martingales, and stochastic calculus. (2016), Springer;
2. I. Karatzas and S. E. Shreve: Brownian motion and stochastic calculus (2nd ed), Springer;
3. S. E. Shreve: Stochastic calculus for finance I and II, Springer.
4. P. Mörters and Y. Peres: Brownian Motion, Cambridge University Press, Cambridge.
Prerequisite Math 540 Real Analysis I - we will review measure theory topics as needed. Math 541 is also nice to have, but not necessary. Math 561 Probability Theory I - you should be willing to spend time and effort on this background material if necessary.
Grading Policy Biweekly Homework (Due Thursday in class): 50% of the course grade. You are encouraged to work together on the homework, but I ask that you write up your own solutions and turn them in separately. There will be few problems assigned; emphasis will be placed on clear, concise, and coherent writing.

Final Paper or Exam (TBA): 50%. Take home final exam or a 3- to 6-page paper summarizing and analyzing a research article that is both interesting to you and related to probability. The final will be due TBA at 5 PM, preferably by email.

Tentative Timeline

Week Date Due Content






1 T Aug 28 Measure Theory basics. Scribe notes from Math 561.
R Aug 30 Construction of Brownian Motion.
Due: A written paragraph about yourself and your interests, including an explanation of why you are taking this course, any information you believe would be helpful to me, and any questions you might have.






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