Unless otherwise noted all readings are from Applied Stochastic Analysis by E, Li and Vanden-Eijnden. Other textbooks are given on the home page. Click the Week links for details on what we cover and suggestions on navigating the readings. I will update these the Friday before we cover the material. Remember to refresh the webpage throughout the term!
The exercises listed on the Weekly pages should be completed by the Monday of the following week. They are not collected, but they are essential for exam preparation. It is strongly recommended that you attempt all exercises in the text related to the material we cover. I may add additional exercises while the material is being covered. If you cannot obtain a solution in a reasonable amount of time on your own, you may use ChatGPT for assistance or come to office hours.
| Day | Date | Topics | Readings |
|---|---|---|---|
| W1 | |||
| Tue | Jan 6 | No Class (meet Xhours) | |
| Thu | Jan 8 | Probability warm-up, Discrete-time Markov chains | 3.1 |
| Fri | Jan 9 (Xhours) | stationary distributions, Perron Frobenius and ergodic theorem | 3.2,3.3 |
| W2 | |||
| Tue | Jan 13 | Poisson processes, Continuous-time Markov chains | 3.4,3.5,3.6 |
| Thu | Jan 15 | Embedded chain, Time reversal, detailed balance | 3.8 |
| W3 | |||
| Tue | Jan 20 | a little measure theory, filtrations, Kolmogorov's extension theorem | 1.2,5.1,5.2, D Theorem 5.8 on pg. 261 |
| Thu | Jan 22 | Gaussian processes, Karhunen–Loève theorem | 5.4 |
| W4 | |||
| Tue | Jan 27 | OU Processes as Markovian Gaussian processes | 5.4 |
| Thu | Jan 29 | Review for midterm | |
| Fri | Jan 30 (Xhours) | Midterm | |
| W5 | |||
| Tue | Feb 3 | General Markov Processes, Infinitesimal generators | 5.3 |
| Thu | Feb 5 | Wiener process as a Gaussian process, Invariance principle, Donsker's theorem, Diffusion equation | 6.1,6.3, 6.2, D7.6 (optional) |
| W6 | |||
| Tue | Feb 10 | Stochastic integrals | 7.1,7.2 |
| Thu | Feb 12 | Itô's lemma, SDEs | 7.3 |
| W7 | |||
| Tue | Feb 17 | Other stochastic integrals and numerical schemes | 7.4,7.5 |
| Thu | Feb 19 | The Fokker–Planck equation | 8.1,8.2,8.3 |
| W8 | |||
| Tue | Feb 24 | Time reversal continued, Monte Carlo methods, importance sampling, variance reduction | 4.2,4.3,4.4 |
| Thu | Feb 26 | Feynman–Kac formula, and particle filtering | 8.5,8.6 |
| W9 | |||
| Tue | Mar 3 | Rare events, large deviations | |
| Thu | Mar 5 | Review |