| Instructor | Yassine El Maazouz |
|---|---|
| Course webpage | http://www.yelmaazouz.org/Ma140B/index.html |
| Course code | Ma/ACM/IDS 140 abc |
| Lectures | MWF 14:00–14:55, 187 Linde Hall |
| E-Mail: | maazouz [at] caltech [dot] edu |
| Office | 308 Linde Hall of Mathematics and Physics |
| Office Hours | W 15:05–16:00 Excluding holidays. Please email me if you cannot make any of these times. |
| Prerequisites | Ma108ABC and Ma140A are strongly recommended. |
| Required Text | There is no required text for this course. However, the following will be used as references:
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| Catalog Description | This course sequence begins with an overview of measure theory, followed by topics that include random walks, the strong law of large numbers, the central limit theorem, martingales, Markov chains, characteristic functions, Poisson processes, and Brownian motion. Towards the end, some further topics may be covered, such as stochastic calculus, stochastic differential equations, Gaussian processes, random graphs, Markov chain mixing, random matrix theory, and interacting particle systems. |
| Syllabus |
I plan to cover the following topics (depending on how much progress we make):
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| Homework |
There will be weekly problem sets. Collaboration (between students) on homework is encouraged.
It is important to make sure you understand the solutions yourself, so you are required to write your own solutions separately.
I also strongly recommend that you spend some time attempting the problems yourself first before discussing with someone else.
You are required to name all your collaborators and sources of information you used for each assignment; any such named source may be used.
You can also use any result discussed in the lectures. DO NOT PLAGIARIZE!
Please write clear and legible solutions. It is strongly recomemded to write your solutions in TeX. Homework is due on Fridays at 6pm and is to be submitted on Gradescope. Late homework submissions will not be accepted. |
| Final | There will be a take-home final due on Monday March 17th at 6pm, to be submitted on Gradescope. Late submissions will not be accepted. You are not allowed to collaborate with anyone on the take-home final. |
| Grading | The final grade will be based 70% on weekly problem sets and 30% on the take-home final. The lowest homework grade will be dropped. |
| Teaching Assistant | Ethan Davis |
| Date | Coverd Material | Problem Sets |
|---|---|---|
| Mo. Jan. 06 2025 | Quick review of conditional expectation and martingales in discrete time + some examples | |
| We. Jan. 08 2025 | Class is cancelled due to wildfires. | |
| Fr. Jan. 10 2025 | Class is cancelled due to wildfires. | |
| Mo. Jan. 13 2025 | Upcrossing inequality and almost sure convergence of martingales. | HW1 |
| We. Jan. 15 2025 | Doob's maximal inequaliy and $L^p$ convergence. | |
| Fr. Jan. 17 2025 | Uniform integrability and $L^1$-convergence. | |
| Mo. Jan. 20 2025 | MLK Jr Day. | HW2 |
| We. Jan. 22 2025 | Optional stopping theorems. | |
| Fr. Jan. 24 2025 | Class cancelled. | |
| Mo. Jan. 27 2025 | Construction of Markov chains. | HW3 |
| We. Jan. 29 2025 | The Markov property. | |
| Fr. Jan. 31 2025 | Classification of states. | |
| Mo. Feb. 03 2025 | Invariant measures 1 | HW4 |
| We. Feb. 05 2025 | Invariant measures 2 | |
| Fr. Feb. 07 2025 | Asymptotic behaviour of Markov chains 1 | |
| Mo. Feb. 10 2025 | Asymptotic behaviour of Markov chains 2 | HW5 |
| We. Feb. 12 2025 | Definition and construction of Brownian motion | |
| Fr. Feb. 14 2025 | First properties of Brownian motion | |
| Mo. Feb. 17 2025 | President's day. | HW6 |
| We. Feb. 19 2025 | Brownian motion path properties 1 | |
| Fr. Feb. 21 2025 | Brownian motion path properties 2. | |
| Mo. Feb. 24 2025 | Simple Markov property | HW7 |
| We. Feb. 26 2025 | Stopping times and right-continuous filtrations | |
| Fr. Feb. 28 2025 | Hitting times of Brownian motion | |
| Mo. Mar. 03 2025 | Strong Markov property | HW8 |
| We. Mar. 05 2025 | Reflexion principle | |
| Fr. Mar. 07 2025 | Markovian processes extracted from Brownian motion | |
| Mo. Mar. 10 2025 | Martingales | HW9 |
| We. Mar. 12 2025 | ||
| Mo. Mar. 17 2025 | FINAL |