+ STAC63: Probability and Stochastic Processes II

STAC63S - Probability and Stochastic Processes II (2024)

Announcements

The Monday class is now in HW216.
The midterm is on Feb. 7, 17:00-19:00 and it will be an online test. A copy of the 2022 midterm with solutions can be found here.
The midterm will cover material up to the end of Lecure 3c.
I will have some online office hours on Sunday, February 4 from 2-5pm. I will post a Zoom link here at 3.
Zoom link for Feb. 4: https://utoronto.zoom.us/j/84850242344
Meeting ID: 848 5024 2344 Passcode: 484026
I will hold some online office hours on Tuesday evening Feb. 6 at 9pm. I will post a Zoom link here at 9.
Zoom link for Feb. 6: https://utoronto.zoom.us/j/86818055775
Meeting ID: 868 1805 5775 Passcode: 665137

There is an error in the solution to Exercise 4.5.5 from Lecture 1. The solution should read as follows.

We use the Poisson(5) distribution as an importance sampler.
Note constant exp(-5) needs to be accounted for.

#generate a sample of n from the Poisson(5)
n=100000
x=rpois(n,5)
est=0
s2=0
for(i in 1:n) {
est=est+sin(x[i]**2)
s2=s2+(sin(x[i]**2))**2
}
est=exp(5)*est/n
s2=exp(5)*exp(5)*s2/n-est**2
cat("estimate +- 3 standard errors","\n")
cat(n,est-3*sqrt(s2/n),est,est+3*sqrt(s2/n),"\n")

10000 -42.25446 -39.57449 -36.89451 
1e+05 -39.7936 -38.94073 -38.08785

Exercise II.3 had two typos in it and these have been corrected. The solution is correct but the typos made it hard to follow.
Solutions to the midterm are here.
There was a typo in the Solutions to the midterm in question 4(b). I wrote P(X_1=1)=1/2 and this should have been P(X_1=1)=1/4. I've gone through all the tests and added marks where appropriate.
Final Exam April 17, 2024, 19:00-22:00 in IC200.
Videographer didn't attebd class of Wednesday, March 13 so there is no video of that class We finished Lecture 5c and began Lecture 5d.
Solutions to 2022 final exam.
The final exam covers all the material in Lectures 1-6c but with only a few marks (<=5) on Lecture 6c. I have put up Lectures 6d and 6e but there will be no questions on this material on the exam. The office hours before the exam on April 17 will be online (with the Zoom link posted here) as follows:
April 10 1-3pm
Join Zoom Meeting
https://utoronto.zoom.us/j/89664260184
Meeting ID: 896 6426 0184
Passcode: 654227

April 16 1-3pm
Join Zoom Meeting
https://utoronto.zoom.us/j/81802845727
Meeting ID: 818 0284 5727
Passcode: 756900

April 17 1-3pm.
Join Zoom Meeting
https://utoronto.zoom.us/j/83135124574
Meeting ID: 831 3512 4574
Passcode: 171127
The final exam is open book.
If you think there is an error in a soltion to an Exercise just email me about the question as then I can check it.

Instructor

Professor Michael Evans
Office: IC498
Tel: (416)287-7274
email: mevans@utstat.utoronto.ca

Time and Place

Three hours of lectures per week.

First class is Monday, January 8.

MO 13:00-15:00 in HW216 (changed from AA206)
WE 11:00-12:00 in SW311
The final exam is open book.

Website

http://www.utstat.utoronto.ca/mikevans/stac63/staC632024.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

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

Class Notes

I will post my class notes here. All my lecture notes from a previous year are listed below to start but these will be updated, and new notes possibly added, in 2024. When the notes to a lecture are completed a * will be appended to the lecture number.

Lecture 1 * Monte Carlo, (Jan. 9 I replaced this lecture by deleting a couple of lines from Example 3 and adding page 15), 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. (updated on March 5)
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 modified March 21)
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 Solutions to Exercises in Lecture 6
Lecture 6d * queuing theory, renewal theory
Lecture 6e * continuous-space Markov chains, MCMC