The course is on Canvas
Lecture Notes
Lecture-notes-1
Lecture-notes-2
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Homework
Homework #1
Homework #2
Homework #3
Solution of Homework #3
Homework #4
Solution of Homework #4
Syllabus
Time and Location
M-W-F Period 5 (11:45 AM – 12:35 PM), LIT 219
Office Hours
Monday 10:30 AM – 11:20 AM, Wednesday 10:30 AM – 11:20 AM, or by appointment
Textbook
There is no required text, but the following textbooks are suggested:
- R. Durrett, Probability: Theory and Examples, 5th edition (PDF available on Prof. Durrett’s website)
- P. Morters and Y. Peres, Brownian Motion, Cambridge University Press (PDF available on Prof. Morters’ website)
- D. Khoshnevisan, Probability, Graduate studies in mathematics vol. 80, 2007
Final Exam Date
Wednesday, May 3 (5/03/2023) at 7:30 AM – 9:30 AM
Scope of the course
The aim of the course is to provide students with strong foundations in the area of probability theory. At the end of the course, students will be acquainted with the language of probability and will gain sufficient experience to successfully apply probabilistic tools to most areas of pure and applied sciences.
The course is intended for graduate students as part of their PhD requirement, and for students considering studying probability theory at a research level.
Prerequisite
MAP 6472 – Probability and Potential Theory I
Topics Covered
Topics include Conditional Expectation, Martingale, Stopping time, Uniform Integrability, Continuous time stochastic processes (Poisson process, Gaussian process, Brownian motion), Potentials and excessive functions. Below is the weekly schedule:
W1: Review of probability (random enumeration of random variables, sigma-algebra generated by random variables, Lebesgue integral, Lebesgue theorems, etc.).
W2: Conditional expectation, construction in L^2, properties of conditional expectation.
W3: Conditional expectation in L^1, independence.
W4: Conditional expectation with respect to random variables, special case: discrete/continuous.
W5: Martingale, Doob-Meyer decomposition.
W6: Stopping time, Doob’s inequality.
W7: Stopped martingale, stopping theorems.
W8: Convergence of martingales, uniform integrability.
W9: Poisson process.
W10: Brownian motion, Levy’s construction of the Brownian motion.
W11: Blumenthal’s 0-1 law, law of large numbers for Brownian motion, long-term behavior, nowhere differentiability of the trajectories.
W12: The Brownian motion as a Gaussian process, as a martingale, as a Markov process.
W13: The reflexion principle.
W14: Potentials and excessive functions.