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
• Feb. 20 - Data Fest Applications are now open until March 1. Please apply here: ASA DataFest Application 2026 – Fill out form [https://forms.office.com/r/DC72t8S75p] . The event takes place in-person (St George campus) May 1-3. Top four teams win cash prizes, with the top prize being $300/team member. There will be industry professionals, and graduate school reps! It's a data-fun-filled weekend
• Feb. 21 - Final exam, April 14 7-10pm in IA2010
• March 3 - I uploaded a new version of Lecture 5d. The stock option example is now optional and the material on conditional expectations is mandatory (there will be a question on this in the final).
• March 7 - Here are the solutions to the midterm. You can pick up your midterm from me after class. I've added 5 marks to everybody which brings the average mark to 65. Just a reminder that the final mark in the course is determined as the maximum of (40% midterm + 60% final exam ) or final exam mark.
• March 12 - I uploaded a slightly changed version of Lecture 6b. This fixed the problem I had in class where I made the mistake of trying to show T_i is a stopping time when i=2. Actually it isn't a stopping time rather the arrival times S_i are stopping times (as now shown in the notes) but there is no reason for the interarrival times to be stopping times.

Instructor

Time and Place

First class is Monday, January 5.

• MO 13:00-15:00 in IA 2010
• WE 11:00-12:00 in IA 2050

Website

Videos

Office Hours

Course Description

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

Textbook

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

• A midterm worth 40% (2 hours probably around Feb. 14).
• A final worth 60% (in-person 3 hours).

Class Notes

• 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. ( Exercise on Conditional Expectation)
  

• Lecture 5b more optional stopping results
  

• Lecture 5c Wald's Theorem
  

• Lecture 5d Martingale convergence theorem, branching process, stock options, Solutions to Exercises (V.1-V.7, Solutions to Exercises (V.8-V.16)
  

• 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