Meeting time: MWF 9:05-9:55 am
Location: Malott Hall 406
Instructor: Daniel Jerison
Office: Malott Hall 581
Office hours: W 10 am - 12 pm, Malott Hall 210 Extra office hours: Friday, May 13, 1-3 pm, Malott Hall 210; Tuesday, May 17, 1-3 pm, Malott Hall 581
Email: jerison at math.cornell.edu
TA: Xiaoyun Quan
Office hours: Th 3-5 pm, Malott Hall 218
Email: xq44 at cornell.edu
Prerequisite: An introductory probability course such as MATH 4710, BTRY 4080, ORIE 3600, ECON 3190. General proficiency in calculus and linear algebra. If unsure about your preparation, please discuss it with me.
Three extra topics not covered this semester are renewal processes, queueing theory, and the connection between random walks and electrical networks. If you are interested to learn more about renewal processes and queueing theory, check Chapter 3 and sections 4.5-4.6 of the textbook. For random walks and electrical networks, there is a very nice introduction in Chapter 9 of Markov chains and mixing times by Levin, Peres, and Wilmer, and much more information in Probability on trees and networks by Lyons and Peres.
This course is modeled after the one taught by Lionel Levine in spring 2013; see that year's course webpage.
Essentials of Stochastic Processes by Durrett, 2nd edition. Available online at this link. Errata.
40%: Weekly homework
15%: In-class prelim
15%: Project
30%: Final exam
The prelim was on Friday, March 11, in class. Prelim and solutions.
The prelim covers all the lecture material through Friday, March 4. This corresponds to sections 1.1-1.9 of the textbook. It is a closed-book, closed-notes exam; no calculators or similar aids are allowed. You are not expected to remember the details of proofs, but you should know the basic ideas. For the Metropolis algorithm, you should know the formula in the case where the proposal transition matrix Q is symmetric, that is, Q(x,y) = Q(y,x). For practice, see the 2013 prelim and 2015 prelim.
The final exam is on Monday, May 16, from 9-11:30 am in Malott Hall 406. It is a closed-book, closed-notes exam; no calculators or similar aids are allowed. Barring the 24-hour rule, you must take the final exam at this time.
Practice final and solutions.
Topics covered on the final exam:
1. All the material that was covered on the prelim, plus Section 1.10 (infinite state space Markov chains, branching processes) excluding Example 1.54.
2. All of Chapter 2: Poisson processes, except there will be nothing about nonhomogeneous Poisson processes.
3. All of Chapter 5: Martingales, except: Lemmas 5.2 and 5.6-5.8; Section 5.4 from Theorem 5.15 to the end; Section 5.5 from Example 5.15 to the end.
4. All of Chapter 6: Mathematical finance, except: Section 6.4 and the Black-Scholes PDE subsection in Section 6.6. You are not expected to memorize the Black-Scholes formula for European call options, but Theorem 6.10 is fair game.
5. Brownian motion: Only the definition, as given in Section 6.6 or the very beginning of the Brownian motion textbook.
Topics NOT covered on the final exam: the English language Markov chain, the Metropolis algorithm formula when the proposal transition matrix is not symmetric, Google PageRank, continuous time Markov chains, card-shuffling and mixing times, modern portfolio theory and the Capital Asset Pricing Model, construction and further properties of Brownian motion.
You are encouraged to work with each other on the weekly homework. Everything you write should be in your own individual words; direct copying is forbidden! If you work in a group, please list the group members at the top of your turned-in homework. You are not allowed to get help from any other person or source on an exam, including the textbook, unless that exam's instructions specifically permit it.
To be continued and subject to change.
Week 1 (1/27, 1/29): Introduction and sections 1.1-1.2.
Week 2 (2/1, 2/3, 2/5): Section 1.3, start of 1.4. Supplement: Proof of Theorem 1.7.
Week 3 (2/8, 2/10, 2/12): Section 1.4, start of 1.5. Supplement: Number theory for Lemma 1.16.
Week 4 (2/17, 2/19): Sections 1.5 and 1.7.
Week 5 (2/22, 2/24, 2/26): Finish sections 1.5 and 1.7. Section 1.6, English language as a Markov chain. English language data.
Week 6 (2/29, 3/2, 3/4): Finish section 1.6, another English language Markov chain. Sections 1.8-1.9. The Markov chain Monte Carlo revolution (first 3 pages only). Supplement: The Metropolis algorithm.
Week 7 (3/7, 3/9, 3/11): Section 1.10. Prelim on Friday, March 11.
Week 8 (3/14, 3/16, 3/18): Google PageRank. Section 2.1, start of 2.2. PageRank papers.
Week 9 (3/21, 3/23, 3/25): Sections 2.2-2.4. What does randomness look like?
Spring break
Week 10 (4/4, 4/6, 4/8): Parts of sections 4.1-4.2, sections 5.1 and start of 5.2.
Week 11 (4/11, 4/13, 4/15): Finish section 5.2, sections 5.3-5.4.
Week 12 (4/18, 4/20, 4/22): Sections 5.5 and start of 6.1. Card-shuffling and mixing times. For more on mixing times, see Markov chains and mixing times by Levin, Peres, and Wilmer. Card-shuffling is covered in Chapter 8.
Week 13 (4/25, 4/27, 4/29): Finish section 6.1, sections 6.2-6.3 and 6.5. Supplement: American call options.
Week 14 (5/2, 5/4, 5/6): Capital Asset Pricing Model, sections 6.6-6.7. For an explanation of the CAPM, first read The Capital Asset Pricing Model: Theory and Evidence by Fama and French from the beginning (page 25) up to the middle of page 29, and then for the mathematical details see Section 1.1 of these notes by Karl Sigman. For a sense of how the CAPM is regarded today, see the previously linked article by Fama and French for a pessimistic view and this interview of William Sharpe for a qualified defense.
Week 15 (5/9, 5/11): Brownian motion. We are covering selected topics from Brownian Motion by Mörters and Peres. The definition and construction of Brownian motion are in Section 1.1, and the strong Markov property and reflection principle are in Section 2.2. The proof that the zero set is uncountable is in Sections 2.1 and 2.2.
Two times during the semester, you may have an automatic 72-hour extension on a problem set. This does not require prior approval, just hand it in at the beginning of the next class with a note saying you are using an extension. Otherwise, late homework will not be accepted except in an emergency (in which case you must inform me as soon as possible).
HW 1: Due Friday, February 5 in class. Solutions.
HW 2: Due Friday, February 12 in class. Solutions.
HW 3: Due Friday, February 19 in class. Solutions.
HW 4: Due Friday, February 26 in class. Solutions.
HW 5: Due Friday, March 4 in class. Solutions.
HW 6: Due Friday, March 18 in class. Solutions to textbook exercises and additional problem.
HW 7: Due Friday, March 25 in class. Solutions.
HW 8: Due Friday, April 8 in class. Solutions to required problems and extra credit problem.
First draft of project: Due Friday, April 15 in class.
HW 9: Due Friday, April 22 in class. Solutions.
Final draft of project: Due Monday, May 2 in class.