Course Description

In this course, we will study several topics in the theory of stochastic processes. Rather than covering many processes briefly, I will focus on a subset of important ones and will try to explore them in depth. Some possible topics include Markov processes, Brownian motion and Stochastic Analysis. I am open to suggestions and may modify the list of topics as the semester progresses.

I will not assume any knowledge of measure theory, but may introduce some basic measure-theoretic concepts along the way.

Full Course Description

Lectures:

1. Lecture 1: Introduction to Markov Chains (1/10/22)

2. Lecture 2: Recurrence Theorem (1/12/22)

3. Lecture 3: Classification Theorem (1/14/22)

4. Lecture 4: Aperiodicity (1/19/22)

5. Lecture 5: Markov Chain Convergence (1/21/22)

6. Lecture 6: Proof of Markov Chain Convergence (1/24/22)

7. Lecture 7: Existence of Stationary Distributions (1/26/22)

8. Lecture 8: Quantitative Convergence Theorem (1/28/22)

9. Lecture 9: Proof of Convergence Theorem (1/31/22)

10. Lecture 10: Upper Bounds on Mixing Times (2/4/22)

11. Lecture 11: Conditional Probability (2/7/22)

12. Lecture 12: Introduction to Martingales (2/9/22)

13. Lecture 13: Optional Stopping Time Theorem and Wald's Identity (2/11/22)

14. Lecture 14: Martingale Convergence Theorem (2/14/22)

15. Lecture 15: Intro to Brownian Motion (2/18/22)

16. Lecture 16: Properties of Gaussians and Brownian Motion (2/21/22)

17. Lecture 17: Invariance and Path-wise Properties of Brownian Motion (2/23/22)

18. Lecture 18: Reflection Principle (2/25/22)

19. Lecture 19: Continuous Martingales (2/28/22)

20. Lecture 20: Continuous Martingales II (3/2/22)

21. Lecture 21: Variation of Brownian Motion (3/4/22)

22. Lecture 22: Motivating the Stochastic Integral (3/7/22)

23. Lecture 23: Constructing the Itô Integral (3/9/22)

24. Lecture 24: The Stochastic Integral as a Process (3/11/22)

25. Lecture 25:

26. Lecture 26: Proof of Itô's Lemma (3/28/22) (Reference: Øskendal 4.1-4.2)

27. Lecture 27: Martingale Representation Theorem (3/30/22) (Reference: Øskendal 4.3)

28. Lecture 28: Intro to Stochastic Differential Equations (4/1/22) (Reference: Øskendal 5.1)

29. Lecture 29: Existence and Uniqueness (4/4/22) (Reference: Øskendal 5.2)

30. Lecture 30: Solving Examples of SDE's (4/6/22)

31. Lecture 31: Diffusions and the Dynkin Formula (4/8/22) (Reference: Øskendal 7.1-7.4)

32. Lecture 32: Applications of Dynkin's Formula (4/11/22)

33. Lecture 33: Girsanov Theory (4/13/22)

34. Lecture 34: Interpolating Processes, Gaussian Concentration and Dimension Reduction (4/15/22)

Homework:

1. Homework 1 (due 2/11/22)

2. Homework 2 (due 3/2/22)

3. Homework 3 (due 3/18/22)

Homework Solution Sketches:

1. Homework 1 Solution Sketch

2. Homework 2 Solution Sketch

Suggested Literature:

Markov Chains

1. Markov Chains and Mixing Times by Levin and Peres

2. A First Look at Stochastic Processes by Rosenthal

3. Adventures in Stochastic Processes by Resnick

Discrete Martingales

1. Probability with Martingales by Williams

2. A First Look at Rigorous Probability Theory by Rosenthal

Brownian Motion, Stochastic Calculus and Stochastic Differential Equations

1. Brownian Motion and Stochastic Calculus by Karatzas and Shreve

2. Brownian Motion by Mörters and Peres

3. Sotchastic Calculus and Financial Application by Steele

4. Stochastic Differential Equations by Øskendal