Probability Theory II
This course is an advanced graduate-level course designed to deepen students’ understanding of the theoretical underpinnings of probability. Building on the foundational concepts introduced in Probability Theory I, this course delves into more complex topics such as martingales, Markov chains, Brownian motion, and stochastic calculus. Through rigorous mathematical proofs and problem-solving exercises, students will explore the interplay between theory and application, gaining insights into how probability theory informs various fields such as finance, physics, and statistics. The course aims to equip students with a solid theoretical background and analytical skills necessary for research and application in their respective disciplines.
Coordinates #
Time: MW 1:30PM - 2:45PM
Location: Maryland 217
Personnel #
Instructor:
Mateo Díaz (mateodd at jhu dot edu)
OH M 3:00PM - 5:00PM Wyman S429
New OH W 3:00PM - 5:00PM Wyman S429
Teaching Assistant:
Ao Sun (asun17 at jhu dot edu)
OH F 9:00AM - 11:00AM Wyman S425
Syllabus #
The syllabus can be found here.
Lecture notes #
Handwritten lecture notes will be posted here and on Canvas.
- Lecture 1: Logistics and motivation
- Lecture 2: Characteristic functions and Levy’s inversion formula
- Lecture 3: Weak convergence and Levy’s Convergence Theorem
- Lecture 4: Poisson distribution and the Law of Rare Events Demo
- Lecture 5: Conditional Expectation
- Lecture 6: Properties of the Conditional Expectation
- Lecture 7: Martingales
- Lecture 8: Doob’s Optional Stopping and Convergence Theorems
- Lecture 9: Examples: Random Walks and Branching Process
- Lecture 10: Convergence in Lp
- Lecture 11: Uniform integrability and Convergence in L1
- Lecture 12: Optional stopping via UI and Backwards martingales
- Lecture 13: Examples: Ballot Theorem, Strong Law of Large Numbers, and de Finetti’s Theorem
- Lecture 14: Markov Chains
- Lecture 15: Markov Property
- Lecture 16: Strong Markov Property
- Lecture 17: Recurrence and Transience
- Lecture 18: Stationary measures
- Lecture 19: Aperiodicity and convergence
- Lecture 20: Brownian motion
- Lecture 21: Existence of Brownian motion
- Lecture 22: The Markov Property for Brownian motion
- Lecture 23: The Strong Markov Property for Brownian motion
- Lecture 24: Brownian motion as a martingale
Textbook #
We will use the following references:
- (Main textbook) Rick Durrett, Probability: Theory and Examples, 5th edition. Cambridge University Press (2019). Available at https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf.
- Patrick Billingsley, Probability and Measure, Anniversary (or 3rd) edition. John Wiley & Sons (2012). Available online through JHU libraries: https://ebookcentral.proquest.com/lib/jhu/detail.action?docID=836625.
- Kai Lai Chung, A Course in Probability Theory. Academic Press (2001).
- David Williams, Probability with Martingales, 1st Edition. Cambridge Mathematical Textbooks (1991).
Grading system #
Your grade will take into account four components: Homework (40%), Take-home exam (25%), Final project (25%), and Participation (10%). In what follows we ellaborate on each of these components.
Homework #
Homework assignments (approximately five) will be posted here and on the course Canvas. Some homework assignments include at least one question that involves the writing and testing of code; Python is prefered. Please submit homework assignments on Gradescope.
Take-home exam #
There will be one take-home exam with a date TBA. The exam will be posted on Canvas, and you will have two days to turn in your solutions through Gradescope. You may not discuss the exam with anyone or otherwise seek external help.
Final project #
There will be a final project; this is an opportunity for students to learn about related topics that we did not cover in class. Students can work in groups of two or three people. The final project includes two deliverables: a written report and a one-hour presentation. Topics for the final project will be released two weeks before the end of class (students are welcome to pick a different project provided the approval of the instructor). Each group can choose how to split the time among the group members. The group should aim to state one result/idea, place it in context (why is it interesting? What implications does it have? Does it open interesting new research directions?), and present the main idea of its proof.
Participation #
Participation weights 10% in the final grade. Engaging in class, Piazza, and office hours will count toward participation. This includes asking questions (even if you think they are silly!) and pointing out typos or mistakes.