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課程名稱 |
機率論二
Probability Theory (Ⅱ) |
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開課學期 |
111-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
林偉傑 |
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課號 |
MATH7510 |
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課程識別碼 |
221EU3420 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
必修 |
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上課時間 |
星期二6,7(13:20~15:10)星期四7(14:20~15:10) |
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上課地點 |
天數102天數102 |
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備註 |
本課程以英語授課。
總人數上限：40人 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
This course is the second-half of a yearly sequence of measure-theoretic probability theory. Tentative topics include: Conditional expectations, martingales, Markov chains, ergodic theory, Brownian motion. If time allows, we will also talk a little bit of stochastic calculus. |
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課程目標 |
Students will understand the basics of measure-theoretic probability. |
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課程要求 |
Probability theory (I). |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
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參考書目 |
Here are some reference books:
Probability and Measure, by P. Billingsley, 3rd edition, Wiley, 1995.
Probability: Theory and Examples, by R. Durrett. 5th edition. Cambridge U. Press 2019.
Probability with martingales, by D. Williams. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991.
Probability theory lecture notes by D. Panchenko. |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Homework |
60% |
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2. |
Exam(s) |
40% |
Midterm 4/18; final exam will be take-home |
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週次 |
日期 |
單元主題 |
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第1週 |
2/21, 2/23 |
2/21: Conditional expectation, conditional probability
2/23: Regular conditional probability |
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第2週 |
3/2 |
3/2: Examples and applications |
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第3週 |
3/7, 3/9 |
3/7: Martingales, branching processes, stopping time, stopped sigma-algebra
3/9: Optional stopping theorem, Doob's maximal inequality |
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第4週 |
3/14, 3/16 |
3/14: Up-crossing inequality, martingale convergence theorem, Doob's martingale, backwards martingales, Hewitt-Savage 0-1 law
3/16: Exchangeability, de Finetti's theorem |
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第5週 |
3/21, 3/23 |
3/21: Proof of the strong law of large numbers using backwards martingales, Markov chains, transition probability, existence
3/23: Examples, Markov property |
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第6週 |
3/28, 3/30 |
3/28: Strong Markov property, recurrence and transience, random walks on Z and Z^2
3/30: Polya's theorem, stationary measure |
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第7週 |
4/6 |
4/6: Perron-Frobenius theorem, existence of stationary measure for an irreducible and recurrent Markov chain |
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第8週 |
4/11, 4/13 |
4/11: Positive and null recurrence, reversible Markov chains, convergence of Markov chains
4/13: Periodicity, total variantion distance, coupling, convergence theorem |
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第9週 |
4/18, 4/20 |
4/18: Midterm
4/20: Stationary sequence, measure-preserving map, ergodicity, examples |
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第10週 |
4/25, 4/27 |
4/25: Birkhoff's ergodic theorem, examples
4/27: Kingman's ergodic theorem, examples |
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第11週 |
5/2, 5/4 |
5/2: Continuous-time stochastic processes, product spaces, Kolmogorov consistency theorem, Brownian motion, lack of continuity
5/4: Continuous modification, Kolmogorov continuity theorem, properties and transformations of Brownian motion |
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第12週 |
5/9, 5/11 |
5/9: Time inversion, nowhere differentiability, stopping times, Markov property
5/11: Strong Markov property, zero set of a Brownian motion, maximum of a Brownian motion |
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第13週 |
5/16, 5/18 |
5/16: Reflection principle, Donsker's theorem, law of the iterated logarithms
5/18: Stochastic integration, simple processes and their stochastic integrals, Ito isometry |
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第14週 |
5/23, 5/25 |
5/23: Stochastic integral for general processes, Ito's formula, examples
5/25: Proof of Ito's formula, high-dimensional Brownian motion, hitting time |
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第15週 |
5/30, 6/1 |
5/30: Recurrence and transience of Brownian motion, Liouville's theorem, the Dirichlet problem, heat equation on the whole space
6/1: Area of two-dimensional Brownian path |
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