STA 447/2006: Stochastic Processes
Wenlong Mou, Department of Statistical Sciences, University of Toronto, Winter 2025
Schedule
| lecture | topic | book chapteres | notes |
| 01/08 | introduction, Markov definitions and transition probabilities, recurrent and transient states, recurrent state theorem | Rosenthal 1.1 – 1.5 | Lecture 1 |
| 01/15 | recurrence of multi-dimensional random walks, f-expansion, gambler's ruin, communicating states and irreducibility, recurrence equivalence theorems | Rosenthal 1.5 – 1.7 | Lecture 2 |
| 01/22 | reducible Markov chains, stationary distribution, reversible Markov chains, criteria for non-existence of stationary distribution, periodicity | Rosenthal 2.1 – 2.3, Lawler 1.5 | Lecture 3 |
| 01/29 | Markov chain convergence theorem, stationary measure of recurrent Markov chains, mean recurrence time and positive recurrence, strong law of large numbers for Markov chains | Rosenthal 2.4, 2.5, 2.8, Durrett 1.7 | Lecture 4 |
| 02/05 | Midterm exam #1, additional topics: strong law of large numbers for Markov chains, applications to MCMC | Durrett 1.7, Rosenthal 2.6 | Lecture 5, Solutions |
| 02/12 | Martingales and stopping times, optional stopping theorems and their proofs, application to gambler's ruin problem | Rosenthal 3.1, 3.2, Lawler 5.3 | Lecture 6 |
| 02/26 | Uniform integrability, Wald's theorem, Doob's maximal inequality, martingale convergence theorem and upcrossing inequality | Rosenthal 3.3, Lawler 5.4 – 5.6 | Lecture 7 |
| 03/05 | L^1 and L^p convergence of martingales, Brownian motion definitions and basic properties, continuous-time OST and applications, reflection principle | Rosenthal 4.1, Lawler 5.5, 5.6, 8.2, 8.3 | Lecture 8 |
| 03/12 | Midterm exam #2, additional topics: branching process, Doob's martingale with applications to posterior distributions | Rosenthal 3.6, Lawler 5.5 | Lecture 9, Solutions |
| 03/19 | stochastic integration, Itô's formula, quadratic variation | Lawler 9.1, 9.2, 9.3 | Lecture 10 |
| 03/26 | generalization of Itô's formula, applications to PDE problems | Lawler 9.4, 9.7, 8.4, 8.5 | Lecture 11 |
| 04/02 | Poisson process and Poisson point processes, continuous-time discrete-space Markov chains | Rosenthal 4.3, 4.4 | Lecture 12 |
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Recommended practice questions
Week 1: Rosenthal 1.3.9, 1.4.8, 1.5.4, 1.5.9
Week 2: Rosenthal 1.5.13, 1.5.14, 1.6.20, 1.6.21, 1.6.25, 1.7.12
Week 3: Rosenthal 2.1.3, 2.4.13, 2.4.14, 2.4.15, 2.4.16, Lawler 1.7, Durrett 1.72
Week 4: Rosenthal 2.4.18, 2.4.20, Durrett 1.70, 1.73, 1.74, 1.75, (if interested, there is a book on Markov chain convergence speeds)
Week 5: (midterm exam 1)
Week 6: Rosenthal 3.1.1, 3.2.3, 3.2.8, 3.2.9, 3.2.10, 3.2.11, Lawler 5.4, 5.8, 5.9 (correction)
Week 7: Rosenthal 3.5.6, 3.5.7, Lawler 5.13, 5.15, Durrett 5.2, 5.6
Week 8: Durrett 5.2, 5.6, 5.9, Lawler 8.8, 8.12, 8.13, Rosenthal 4.1.12, 4.1.13
Week 9: (midterm exam 2)
Week 10: Lawler 9.1, 9.2, 9.6, additional questions
Week 11: Lawler 9.3, additional questions
Week 12: Rosenthal 4.3.17, 4.3.18, 4.4.14, 4.4.16, Durrett 2.29
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