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STAC63: Probability and Stochastic Processes II
STAC63S - Probability and Stochastic Processes II (2026)
Announcements
- Jan. 15 - Midterm date, time and place - February 27, 2026 15:00-17:00 in IA2021
- Feb. 8 - The midterm on Feb. 27 will cover up to the end of Lecture 4d.
- Feb. 8 - I won't be back in class until after Reading Week but my plan is to offer some
online office hours during Reading Week. I will post times and links here.
- Feb. 10 - Midterm 2024 Solutions
Instructor
Professor Michael Evans
Office: IA4026
email: mevansthree.evans@utoronto.ca
Time and Place
Three hours of lectures per week.
First class is Monday, January 5.
- MO 13:00-15:00 in IA 2010
- WE 11:00-12:00 in IA 2050
Website
http://www.utstat.utoronto.ca/mikevans/stac63/staC632025.html
Videos
There will be videos of the classes placed up on Quercus.
Office Hours
The in-person office hours will be right after class on Wednesdays in my office for 3 hours.
Course Description
STAC63S is a theoretical course but with more emphasis on applications than STAC62F. Note that STAC62 is a prerequisite of STAC63 (Exclusions STA447H1, STA348H5).
The following topics will be covered. The material on Monte Carlo will be interspersed throughout the course.
- Monte Carlo methods
- A mini review of some things covered in STAC62 including proofs of some results not proved in STAC62.
- Markov chains
- Martingales
- Continuous processes - Poisson process, Brownian motion, renewal and queueing theory
There will be Exercises assigned, often from the book, and solutions subsequently provided. It is important that
you do the Exercises to prepare for the midterm and final.
Textbook
A First Look at Stochastic Processes by Jeffrey S. Rosenthal published by World Scientific
Other Sources
- Grimmett and Stirzaker (2001) Probability and Random Processes, Third Edition, Oxford.
- Arguin (2022) - A First Course in Stochastic Calculus, American Mathematical Society.
Evaluation
The midterm and final are open book.
- A midterm worth 40% (2 hours probably around Feb. 14).
- A final worth 60% (in-person 3 hours).
Class Notes
I will post my class notes here.
- Lecture 1 Monte Carlo,
Solutions to Exercises
- Lecture 2 Convergence - modes of convergence, continuous mapping theorems, the delta theorem,
Solutions to Exercises
- Lecture 3a Markov Chains - basic definitions and results, read Chapter 1 in the text.
- Lecture 3b Transience and recurrence, irreducibility
- Lecture 3c Gambler's ruin
Solutions to Exercises in Lecture 3
- Lecture 4a Stationary distributions, time reversibility, read Chapter 2 in the text.
- Lecture 4b Period of a state, Markov Chain Convergence Theorem
- Lecture 4c Markov Chain Monte Carlo
- Lecture 4d Random walks on graphs, positive and null recurrence,
Solutions to Exercises in Lecture 4
- Lecture 5a review conditional expectation, martingales, stopping times, read Chapter 3 in the text.
- Lecture 5b more optional stopping results
- Lecture 5c Wald's Theorem
- Lecture 5d Martingale convergence theorem, branching process, stock options,
- Lecture 6a Brownian motion, read Chapter 4 in the text, Simulated Brownian motion plot, Plot of inverse Gaussian density with lambda = b^2, mu = infinity
- Lecture 6b Poisson processes
- Lecture 6c Continuous time discrete state space processes
- Lecture 6d queuing theory, renewal theory
- Lecture 6e continuous-space Markov chains, MCMC