STA 447/2006: Stochastic Processes
Wenlong Mou, Department of Statistical Sciences, University of Toronto, Winter 2024
Schedule
| lecture | topic | book chapteres | notes |
| 01/10 | introduction, Markov chain definitions, multi-step transition | 1.1 – 1.3 | Lecture 1 |
| 01/12 | classification of states, hitting time, recurrent state theorem | 1.4, 1.5 | Lecture 2 |
| 01/17 | gambler's ruin, communicating states and irreducibility | 1.7, 1.6 | Lecture 3 |
| 01/19 | recurrence of finite and infinite-state markov chains, applications | 1.6 | Lecture 4 |
| 01/24 | stationary distribution, reversible Markov chains | 2.1, 2.2 | Lecture 5 |
| 01/26 | connection between stationarity and recurrence, periodicity, convergence to stationary | 2.2 – 2.4 | Lecture 6 |
| 01/31 | proof of Markov chain convergence theorem | 2.4 | Lecture 7 |
| 02/02 | average convergence for periodic chains, mean recurrence time, stationary measure | 2.5, 2.8, Durrett 1.7 | Lecture 8 |
| 02/07 | proof of recurrence time theorem, convergence of test functionals | Durrett 1.7 | Lecture 9 |
| 02/09 | applications of Markov chain theory, martingale definitions, stopping time | 2.6, 2.7, 2.9, 3.1, 3.2 | Lecture 10 |
| 02/14 | optional stopping theorems | 3.2 | Lecture 11 |
| 02/16 | midterm exam #1 | - | Solutions |
| 02/28 | applications of optional stopping theorems | 3.2, Lawler 5.3 | Lecture 12 |
| 03/01 | uniform integrability, Wald's theorem, martingale convergence | 3.3, Lawler 5.5 | Lecture 13 |
| 03/06 | examples of martingale convergence | Lawler 5.5 | Lecture 14 |
| 03/08 | more examples; Brownian motion definitions, reflection principle | 3.6, 4.1, Lawler 8.2 | Lecture 15 |
| 03/13 | more on reflection principle, Brownian motion as a martingale, scaling and zero sets | 4.1, Lawler 8.3 | Lecture 16 |
| 03/15 | stochastic integration, It̫'s calculus | Lawler 9.2 Р9.4 | Lecture 17 |
| 03/20 | applications of stochastic integration | Lawler 8.4, 9.4, 9.5 | Lecture 18 |
| 03/22 | midterm exam #2 | - | Solutions |
| 03/27 | Poisson processes | 4.3 | Lecture 19 |
| 04/03 | properties of Poisson processes | 4.3 | Lecture 20 |
| 04/05 | continuous-time discrete-space Markov processes | 4.4 | Lecture 21 |
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Recommended practice questions
Week 1: 1.3.9, 1.4.8, 1.5.9, 1.5.13, 1.5.14
Week 2: 1.6.20, 1.6.21, 1.6.25, 1.7.12, 1.7.13
Week 3: 2.1.3, 2.4.13, 2.4.14, 2.4.15, 2.4.16
Week 4: 2.4.18, 2.4.20, 2.5.6, 2.5.7, Durrett 1.72
Week 5: 2.6.6, 2.9.3, Durrett 1.70, 1.73, 1.75
Week 6: 3.2.3, 3.2.8, 3.2.9, 3.2.10, 3.2.11
Week 7: 3.5.7, Lawler 5.4, 5.8, 5.9 (correction), 5.15
Week 8: 3.5.6, 4.1.6, Lawler 5.11, 8.4, 8.11 (for calculation questions, you don't need to compute the numerical value, giving the formulae in the forms of integrals is enough.)
Week 9: Lawler 8.8, 8.12, 9.3, additional exercises (solutions)
Week 10: Lawler 9.1, 9.2, 9.6, additional exercises (solutions)
Week 11: 4.3.6, 4.3.7, 4.3.8, Durrett 2.22, 2.29
Week 12: 4.3.17, 4.3.18, 4.4.14, 4.4.16, Durrett 2.52
Final week: Practice questions, Solutions, (Additional Poisson questions: Durrett 2.59, 2.61)
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