Math 562: Theory of Probability II (Fall 2021)

                  SYLLABUS                COMPASS2G                  DISCUSSION BOARD                

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.

Instructor Partha Dey
Office341A Illini Hall
ContactBy email with subject line: "Math 562:"
Class TR 9:30am -10:50am in 207 Psychology Building.
DiscussWrite your question, discuss any topic on campuswire
Office Hrs Tuesday after class (11:00am - 11:50am) or By appointment. 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. 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. Here is the lecture note from last semester.
Grading Policy Scribe notes: 10% of the course grade. Scribe the notes from 1 (or 2) lectures using the latex template with complete details.
Biweekly Homework (Due Thursday 9:30am via compass): 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 Exam (TBA): 40%. Take home final exam due TBA, by COMPASS2G.

Week Date Due Content

1 T Aug 24 Introduction to Brownian Motion, source.
R Aug 26 Construction of Pre-Brownian Motion, source.
Due: Aug 27 A 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.

2 T Aug 31 Kolmogorov's Continuity Theorem, source.
R Sep 02 Properties of Brownian Motion (by Anna Winnicki), source.

3 T Sep 07 Properties of Brownian Motion (by Anna Winnicki), source.
R Sep 09 Homework 1 Filtrations and Stopping Times (by Michael Wieck-Sosa), source.

4 T Sep 14 Martingales and Upcrossings (by Michael Wieck-Sosa), source.
R Sep 16 HW1 Soln Martingale Convergence Theorems (by Aditya Gopalan), source.

5 T Sep 21 Strong Markov Property (by Aditya Gopalan), source.
R Sep 23 Homework 2 Finite Variation processes (by Rentian Yao), source.

6 T Sep 28 Continuous Local Martingales (by Rentian Yao), source.
R Sep 30 HW2 Soln Quadratic Variation Process (by David Lundquist), source.

7 T Oct 05 Continuous Semi-martingales (by David Lundquist), source.
R Oct 07 Homework 3 Stochastic Int for L2-Bounded Martingales (by Shi Qiu), source.

8 T Oct 12 Stochastic Int for Semimartingales (by Shi Qiu), source.
R Oct 14 HW3 Soln Itô's formula, (by Xueze Song)source.

9 T Oct 19 Consequences of Itô's formula (by Xueze Song), source.
R Oct 21 Homework 4 BDG inequality (by Xuechao Wang), source.

10 T Oct 26 Martingales as Stochastic Integrals (by Xuechao Wang), source.
R Oct 28 HW4 Soln Girsanov's Theorem (by ), source.

11 T Nov 02
R Nov 04 Homework 5

12 T Nov 09 HW5 Soln
R Nov 11

13 T Nov 16
R Nov 18 Homework 6

14 T Nov 23 No classes. Thanksgiving break.
R Nov 25

15 T Nov 30
R Dec 02 HW6 Soln

16 T Dec 07

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