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南开大学STAT0041随机分析
这里是南开大学统计与数据科学学院为高年级本科生开设的随机分析课程主页。
本课程前置课程为初等概率论，随机过程，不要求修读测度论等高级课程，培养学生利用随机分析的思想对具有随机性的问题进行数学建模，并通过随机分析的方法解决问题的能力。
本课程强调概率直观，注重理论与应用相结合，包含随机分析在前沿生成式人工智能领域的应用选讲。
授课教师：赵尉辰 上课时间：周一3-4 上课地点：二主楼B201 助教信息：江杰
Lecture notes & slides
Lecture 1 - Introduction & Preliminaries [Notes][Slides] Lecture 2 - Conditional Expectation [Notes] Lecture 3 - Discrete-time Martingale [Notes] Lecture 4 - Optional Stopping Theorem [Notes] Lecture 5 - Continuous-time Martingale [Notes][Slides] Lecture 6 - Brownian Motion [Notes] Lecture 7 - Properties of Brownian Motion [Notes] Lecture 8 - Brownian Motion and Heat Semigroup [Notes] Lecture 9 - Itô stochastic integral [Notes] Lecture 10 - Properties of Itô integral [Notes] Lecture 11 - Itô Formula [Notes] Lecture 12 - Stochastic Differential Equations [Notes] Lecture 13 - Diffusion Process [Notes] Lecture 14 - Ornstein–Uhlenbeck Process and Langevin Dynamics [Notes]
Advance Topics
Lecture 15 - Introduction to Sampling and Diffusion Models [Slides] Lecture 16 - Langevin Algorithm [Slides] Lecture 17 - Denoising Diffusion Model [Slides]
Reference books
随机分析引论，钱忠民、应坚刚，复旦大学出版社，ISBN 978-7309125689，2017 随机分析基础（第二版），黄志远，科学出版社，ISBN 978-7030087508，2001 Stochastic differential equations: an introduction with applications, Bernt Øksendal, Springer, ISBN 978-3540637202, 2010 随机微分方程及其应用概要，龚光鲁，清华大学出版社，ISBN 978-7302167761，2008
Homeworks
Homework 1 - Preliminaries [pdf] Homework 2 - Martingale [pdf] Homework 3 - Brownian Motion [pdf] Homework 4 - Ito Integral [pdf] Homework 5 - Stochastic Differential Equations [pdf]
Suggested readings
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