Instructor: Alexander Dunlap, Ph.D. (he/him; dunlap at math decimal duke decimal edu). Office: Physics 297. Office hours: Tuesdays 4:30–5:30pm, Wednesdays 3:00pm–4:30pm (subject to change), or by appointment. Please use email only for administrative issues. If you have questions about course content, please come to office hours or use our Ed Discussions site (accessible via Canvas). I will check emails and Ed Discussions at least once each workday; please do not expect immediate responses (e.g. on weekends or right before the homework is due).
Meetings: Tuesdays (T) and Thursdays (Θ), 11:45am–1:00pm, Physics 130.
Textbook: Gregory Lawler, Introduction to Stochastic Processes, 2nd. ed.
Course description: This course will be an introduction to stochastic (random) processes that often come up in applications. We will discuss Markov chains, branching processes, the Poisson process, martingales, Brownian motion, and stochastic integration. This course will not use measure theory, and thus at times we will not be able to operate at a 100% rigorous level.
Course requirements: There will be homework assignments every week or two; you will have at least a week to complete each assignment. Homework assignments will be submitted on Gradescope. A (possibly proper) subset of the problems will be graded. There will be a midterm exam in class on Thursday, October 12, and a final exam as scheduled by the registrar (currently listed as Friday, December 15, 2pm–5pm). Exam problems will be similar to those on the homework.
Grading: The final grade will be computed as a combination of 20% homework, 30% midterm, 50% final exam. The lowest two homework assignments will be dropped.
(very much subject to change, but I will try to keep this table up-to-date)
| Meetings | Topic | Textbook sections |
|---|---|---|
| T 8/28–Θ 8/30 | Finite Markov chains. Invariant measures. Classification of states. | §1.1–1.3 |
| T 9/5–Θ 9/7 | Classification of states, cont'd. Return times. Transient states. | §1.3–1.6 |
| T 9/12–Θ 9/14 | Countable Markov chains. Recurrence, transience, positive recurrence, null recurrence. | §2.1–2.3 |
| T 9/19–Θ 9/21 | Branching process, Poisson process. | §2.4, §3.1 |
| T 9/26–Θ 9/28 | General continuous-time Markov chains. | §3.2–3.4 |
| T 10/3–Θ 10/5 | Reversibility of Markov chains | Ch. 7 |
| T 10/10 | Optimal stopping of Markov chains | Ch. 4 |
| Θ 10/12 | Midterm exam in class | |
| T 10/17 | Fall break, no class | |
| Θ 10/19 | Conditional expectation | §5.1 |
| T 10/24–Θ 10/26 | Martingales and the optional sampling theorem | §5.2–5.3 |
| 🎃 10/31–Θ 11/2 | Uniform integrability and the martingale convergence theorem | §5.4–5.5 |
| 🗳️ 11/7–Θ 11/9 | Martingale convergence, cont'd. Maximal inequalities. | §5.5–5.6 |
| T 11/14–Θ 11/16 | Review of random walk. Introduction to Brownian motion. Markov property of Brownian motion. | §8.1–8.2 |
| T 11/21 | Brownian motion in higher dimensions. Introduction to path properties of Brownian motion. | §8.4 |
| 🙏 11/23 | Thanksgiving, no class | |
| T 11/28 | Recurrence and transience of Brownian motion. | §8.5 |
| Θ 11/30–Θ 12/7 | Introduction to stochastic integration and Itô's formula. | §9.1–9.3 |