Math 562, Probability Theory II
Fall 2013

Course Information

Professor:Kay Kirkpatrick
Office:334 Illini Hall
Lectures: MWF 11:00-11:50 AM (141 Altgeld Hall)
Office hours: By appointment: for instance, before and after class are good times 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.
Recommended reading: Karatzas and Shreve: Brownian Motion and Stochastic Calculus (2nd edition), Springer, 1991.
Gopen and Swan: The Science of Scientific Writing
Williams: Style: The Basics of Clarity and Grace (any edition), Longman.
Other resources: UIUC Writers Workshop, Purdue Online Writing Lab
Useful courses to have: 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: Homework: 50% of the course grade. Some of the homework will emphasize writing and revising skills.
Final Paper: 50%. A 3- to 6-page paper summarizing and analyzing a research article that is both interesting to you and related to probability. UPDATE (due date): Our final exam period is Friday, 12/20/2013, 8-11 AM, but you are not required to be there; instead the final paper will be due the same day at 5 PM, preferably by email.

Homework (due Fridays in class): 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.

HW #1, due the first Friday in class: one 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.

HW #2 (announced in class), due Fri. Sept. 6: Karatzas and Shreve p. 5, problem 1.1.16.

HW #3, due Fri. Sept. 13: Read John Lee's advice on writing proofs and do problems 2.9 and 2.10 on p. 55.

HW #4, due Fri. Sept. 27: Do problems 5.12 and 5.17 in section 1.5. Plus a writing exercise: pick one of these two problems, and explain the context and the key ideas of the problem--in words, with minimal notation and jargon.

HW #5, due Fri. Oct. 4: Paul Halmos was a famously good mathematical writer. Read sections 0-5 of his How to Write Mathematics essay and do problem 1) 3.2.5, 2) 3.2.27, or 3) 3.2.29 according to your number assigned in class. (If you forget or didn't get your number, generate one for yourself uniformly at random from the set {1, 2, 3}.) You may look at the other two problems, but don't work them out.
P.S. If you do a lot of mathematical writing, you might want to invest in How to Write Mathematics, a book of essays that includes Halmos's.
P.P.S. It's a good time to start learning how to write in LaTeX or some equivalent. You could ask a classmate for previous homework code. And Detexify is a helpful tool.

HW #6, due Fri. Oct. 11: On the previous Friday you will have exchanged homeworks with a classmate who did a different problem. Write a peer review of your classmate's homework after consulting advice here or here, for instance. On Oct. 11, you will turn in both the original problem and the peer review (stapled or clipped together).

HW #7 (Proposal), due Fri. Oct. 18: Start by copying down the full citation(s) including title and abstract of the proposed research article(s) that your final paper will be about. Then write 500-1000 words persuading me that you are the right person to study this article, that the topic is important, and that it is interesting to me. Include a motivating question that your final paper will answer, or a thesis statement. You may include references to secondary papers at the end, and they don't count towards the word limit.

HW #8, due Fri. Nov. 1: Finish reading Halmos's How to Write Mathematics essay (sections 6 and 7 are optional, so just sections 8-20). Then pick about 10-20 lines of text and equations from a probability textbook (e.g., Karatzas and Shreve or Durrett) that you think is not very clear. Copy the passage and then improve it using the principles you've learned: add explanations and clarifications, rearrange the exposition if you think another approach is better, or simplify it to an interesting special case and explain the extension to the general case. As an alternative, you may do the same exercise on a Wikipedia article in probability, taking screenshots (for instance) before and after. In both cases, you should append a couple of sentences about why your revision is better.

Assignment for Fri. Nov. 8 (nothing to hand in): Read Gopen and Swan's piece The Science of Scientific Writing. Then look over your proposal with this advice in mind, because you may want to revise and adapt parts of your proposal for the final paper.

Assignment for Fri. Nov. 15 & 22 (nothing to hand in): Find an electronic or paper copy (library or borrow one of mine) of Joseph Williams' Basics of Clarity and Grace (any edition) and read as much as you can. You may also want to start outlining or drafting your final paper.

HW #9, due Fri. Dec. 6: A draft of your final paper. You're welcome to adapt parts of your proposal; the key word is "adapt," because the purpose of the final paper is to inform, whereas the purpose of the proposal was to persuade.

HW #10, due Fri. Dec. 13: Revising exercise: cutting words. Option 1): take an old email of yours to someone important that was too long (more than 2-3 paragraphs of 2-3 lines each), and trim it down without losing key information. You may fictionalize names, etc., for confidentiality. Include word counts before and after: the after count should be no more than 85% of the before count. Option 2): take the draft of your final paper and trim it down to the 3-6 page limit or to 85% of the previous length. In both cases, you should be able to use what you learned from Williams, Gopen, and Swan, to eliminate unnecessary words and phrases and to make your writing more compact and more elegant.


This is the second half of a 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 martingales; (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 cover some other topics, perhaps from mathematical finance or statistical mechanics.

Why work on your writing?

"I am sorry I have had to write you such a long letter, but I did not have time to write you a short one." --Blaise Pascal

I think that success in your career (any career) will depend on how well you communicate your ideas and persuade other people, so I am giving you a chance to learn and practice good writing skills. Several writing assignments will lead up to the final paper, and homework solutions will be graded partly on clarity, concision, and coherence. This is a wonderful opportunity to improve your writing skills, in order to make your ideas more clear and persuasive--and to succeed.