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課程資訊

課程名稱

機率論二
Probability Theory (Ⅱ)

開課學期

106-2

授課對象

理學院數學研究所

授課教師

王振男

課號

MATH7510

課程識別碼

221 U3420

班次

學分

3.0

全/半年

半年

必/選修

必修

上課時間

星期二6,7(13:20~15:10)星期四7(14:20~15:10)

上課地點

天數302天數302

備註

總人數上限：40人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1062MATH7510

課程簡介影片

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課程概述

This is the second half of the graduate probability course. In this semester, we plan to cover the conditional probability, martingales, Markov processes, Brownian motion.

課程目標

The goal of the course is to introduce the concepts of conditional probability and martingales. Based on that, we will discuss several well known stochastic processes. We hope to equip students sufficient knowledge to get into more advanced probability topics.

課程要求

Attending lectures, Homework assignments, Final exam.

預期每週課後學習時數

Office Hours

另約時間

指定閱讀

Olav Kallenberg, Foundations of Modern Probability, Second Edition, Springer-Verlag.

參考書目

1. Probability: Theory and Examples, by R. Durrett. 4rd edition. Cambridge U.
Press 2010.
2. A Course in Probability Theory, by K.L. Chung, second edition, Academic Press, 1974.
3. Probability and Measure, by P. Billingsley, 3rd edition, Wiley, 1995.

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	作業	70%
2.	期末考	30%

課程進度

週次	日期	單元主題
第1週	2/27,3/01	Sub sigma-field, Conditional expectations, Radon-Nikodym theorem, Properties of conditional expectations, Conditional Markov's inequality, Conditional Jensen's inequality, Conditional Cauchy-Schwartz inequality, Projection operator.
第2週	3/06,3/08	Regular conditional probabilities, Regular condition distributions, Product space case, Existence of regular conditional distributions, Polish space, Disintegration.
第3週	3/13,3/15	No class
第4週	3/20,3/22	Application of regular conditional distribution: conditional Jensen's inequality, conditional Cauchy-Schwarz, Uniform integrability, Conditional independence, Doob's characterization.
第5週	3/27,3/29	Filtrations, Adaptedness, Optional times, $\sigma$-field associated with an optional time, Right-continuous filtrations, Weakly optional times, Closure properties.
第6週	4/03,4/05	Spring break
第7週	4/10,4/12	Discrete approximation of a weakly optional time, Progressive measurability, Optional evaluation, Hitting time, Debut theorem, Completeness, Right-continuous extension, Change of time, Martingale.
第8週	4/17,4/19	Martingales (submartingales, supermartingales), Predictable random sequence, Doob decomposition, Convex transform of a martingale (submartingale), Optional sampling theorem, Martingale criterion, Martingale transform, Optional stopping.
第9週	4/24,4/26	Gambling system, Stopped process, Doob's maximal inequalities, Norm inequalities, Doob's upcorssing lemma.
第10週	5/01,5/03	Doob's convergence theorem, One-sided convergence, Extended Borel-Cantelli's lemma, Closed and closure, Uniform integrability and closure, L^p convergence, Conditional limits.
第11週	5/08,5/10	Tail \sigma-field, Kolmogorov's 0-1 law, Reproof of SLLN, Martingales with continuous parameter, Right-continuous and Left-hand limits (rcll), Backward martingales, Optional sampling and closure.
第12週	5/15,5/17	Optional sampling and closure (continuous parameter), Markov processes, Probability kernels, Transition kernels, Chapman-Kolmogorov relation.
第13週	5/22,5/24	Self-study week
第14週	5/29,5/31	Space-homogeneous transition kernels, Independent increments, Time-homogeneous, Semigroups, Existence of a Markov process, Canonical process, Initial distribution, Strong Markov property.
第15週	6/05,6/07	Proof of strong Markov property, Invariant distributions, Stationary processes, First hitting times, Occupation times, Recurrent and transient, Examples of Markov processes, Markov Chains, Irreducible Markov chains, Recurrence class.
第16週	6/12,6/14	Irreducible Markov chains, Invariant distributions, Ergodic behaviors, Method of coupling.
第17週	6/19,6/21	Independent increments, Stationary increments, Infinitely divisible distributions, Levy-Khintchine formula, Levy processes.