15.070J 高级随机过程与离散概率

MIT Course 15 · Operations Research/Statistics · 课号 15.070J

TL;DR

• 严谨测度论视角的随机过程，深入离散/连续时间马尔可夫链、泊松点过程与鞅论。
• 稳态分布存在性、遍历性定理与首达时分析量化动态随机系统的长期极限行为。
• 随机微分方程与更新理论为金融工程、通信排队与可靠性物理学提供数学公理底座。

课程简介 / Description

This class covers the analysis and modeling of stochastic processes. Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. In addition, the class will go over some applications to finance theory, insurance, queueing and inventory models.

讲义 / Lecture Notes

• 1
  • Metric spaces and topology
  • Lecture 1: Metric spaces (PDF)
• 2
  • Large deviations for i.i.d. random variables
  • Lecture 2: Large deviations technique (PDF)
• 3
  • Large deviations theory
  • Cramér's theorem
  • Lecture 3: Cramér's theorem (PDF)
• 4
  • Applications of the large deviations technique
  • Lecture 4: Applications of large deviations (PDF)
• 5
  • Extension of LD to ℝ
  • d
  • and dependent process
  • Gärtner-Ellis theorem
  • Lecture 5: LD in many dimensions and Markov chains (PDF)
• 6
  • Introduction to Brownian motion
  • Lecture 6: Intro Brownian motion (PDF)
• 7
  • The reflection principle
  • The distribution of the maximum
  • Brownian motion with drift
  • Lecture 7: Brownian motion (PDF)
• 8
  • Quadratic variation property of Brownian motion
  • Lecture 8: Quadratic variation (PDF)
• 9
  • Conditional expectations, filtration and martingales
  • Lecture 9: Filtration and martingales (PDF)
• 10
  • Martingales and stopping times I
  • Lecture 10: Martingales I (PDF)
• 11
  • Martingales and stopping times II
  • Martingale convergence theorem
  • Lecture 11: Martingales II (PDF)
  • Additional materials: Martingale convergence theorem (PDF)
• 12
  • Martingale concentration inequalities and applications
  • Lecture 12: Martigales concentration inequality (PDF)
• 13
  • Concentration inequalities and applications
  • Lecture 13: Talagrand's concentration inequality (PDF)
• 14
  • Introduction to Ito calculus
  • Lecture 14: Ito calculus (PDF)
• 15
  • Ito integral for simple processes
  • Lecture 15: Ito construction (PDF)
  • Midterm Exam
• 16
  • Definition and properties of Ito integral
  • Lecture 16: Ito integral (PDF)
• 17
  • Ito process
  • Ito formula
  • Lecture 17: Ito process and formula (PDF)
• 18
  • Integration with respect to martingales
  • Notes unavailable
• 19
  • Applications of Ito calculus to financial economics
  • Lecture 19: Ito applications (PDF)
• 20
  • Introduction to the theory of weak convergence
  • Lecture 20: Weak convergence (PDF)
• 21
  • Functional law of large numbers
  • Construction of the Wiener measure
  • Lecture 21: Tightness of measures (PDF)
• 22
  • Skorokhod mapping theorem
  • Reflected Brownian motion
  • Lecture 22: Reflected Brownian motion (PDF)
  • Final Exam

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