This is the second course in the graduate probability sequence, and the sequel to STAT 210. The course will cover discrete-time martingale theory, Brownian motion, and Ito calculus.
Staff and Organization
Instructor: Mark Sellke (msellke@fas.harvard.edu)
Teaching Fellow: Somak Laha (somaklaha@fas.harvard.edu)
Lecture Location:
Science Center 705
Sever Hall 103
Important Websites: Canvas
Prerequisites
Probability at the level of STAT 210, and real analysis at the level of MATH 112.
Sections and Office hours
Mark's Office Hours: Wednesday at 10:30am-11:30am and 2pm-3pm (Science Center 711)
Somak's Sections: 4:30pm-5:30pm Tuesday/Thursday
Somak's Office Hours: 5:30pm-6:30pm Tuesday/Thursday
Location: Science Center 304 on Tuesday, and Science Center 705 on Thursday
Course Materials
Some useful books for this course are by Durrett (especially Chapters 4 and 7), Mörters and Peres, Steele, and Revuz and Yor.
Grading
Homework assignments (30%), Midterm (25%), Final (40%), Scribing (5%).
Scribing
Students will signup to scribe during the first week of class. Scribe notes (both PDF and source files) should be emailed to Mark and Somak within 24 hours of the class. This Latex template is recommended but not required (it can be copy-pasted into a new project). If you are not the only scribe for the day, please consolidate your notes into a single version before sending it to us. Scribes are especially encouraged to ask clarifying questions during lectures.
Assignment Schedule
| Assignment | Deadline |
|---|---|
| Homework 1 | 11:59pm ET 2/7/2025 |
| Homework 2 | 11:59pm ET 2/21/2025 |
| Homework 3 | 11:59pm ET 3/7/2025 |
| Midterm | In class, 3/10/2025 (Monday) |
| Homework 4 | 11:59pm ET 4/4/2025 |
| Homework 5 | 11:59pm ET 4/18/2025 |
| Homework 6 | 11:59pm ET 4/30/2025 |
| Extra Credit Problems | 11:59pm ET 4/30/2025 |
| Final Exam | TBA |
Submitting Problem Sets
Problem sets should be submitted through Gradescope, as a PDF file. The PDF may be typed (e.g. in Latex), or neatly hand-written and scanned (please check for legibility in this case). Make sure to select which pages correspond to which problems, to ensure all of your solutions are graded. See this helpful short video for clarification.The lowest homework grade will be dropped, and you may use up to two late days throughout the semester
Course Schedule (Tentative)
| Date | Topic | Scribe Notes |
|---|---|---|
| Jan 27 | Course Overview, Radon-Nikodym Theorem. | Notes |
| Jan 29 | Conditional Expectation and Martingales | Notes |
| Feb 3 | Uniform Integrability | Notes |
| Feb 5 | More on UI, Lp Maximal Inequalities | Notes |
| Feb 10 | Backwards Martingales | Notes |
| Feb 12 | Concentration of Martingales | Notes |
| Feb 17 | (Holiday; no class) | -- |
| Feb 19 | Construction of Brownian Motion | Notes |
| Feb 24 | Roughness of Brownian Motion | Notes |
| Feb 26 | Continuous-Time Filtrations, Blumenthal 0/1 Law | Notes |
| Mar 3 | Strong Markov Property and Reflection Principle | Notes |
| Mar 5 | Brownian Motion as a Continuous Martingale | Notes |
| Mar 10 | Midterm | -- |
| Mar 12 | Convergence to Brownian Motion in Path Space | Notes |
| Mar 17-21 | Spring Break | -- |
| Mar 24 | Prohorov's Theorem and Paley-Wiener Integration | Notes |
| Mar 26 | Progressively Measurable Processes | Notes |
| Mar 31 | Random Matrix Theory (guest lecture by Kevin Yang) | Notes |
| Apr 2 | Construction of Ito Integral | Notes |
| Apr 7 | Ito's lemma | Notes |
| Apr 9 | Stochastic Differential Equations I | Notes |
| Apr 14 | Stochastic Differential Equations II | Notes |
| Apr 16 | Lévy's Characterization of Brownian Motion, Dubins-Schwarz Theorem | Notes |
| Apr 21 | Brownian Motion and Complex Analysis | Notes |
| Apr 23 | Picard's Little Theorem and Cameron-Martin | Notes |
| Apr 28 | Girsanov's Theorem | |
| Apr 30 | Diffusion Sampling | |
| TBA | Final Exam | -- |