Math 6550
Introduction to Stochastic Processes
Spring 2022
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.
Lectures:
Lecture 13: Optional Stopping Time Theorem and Wald's Identity (2/11/22)
Lecture 14: Martingale Convergence Theorem (2/14/22)
Lecture 16: Properties of Gaussians and Brownian Motion (2/21/22)
Lecture 17: Invariance and Path-wise Properties of Brownian Motion (2/23/22)
Lecture 20: Continuous Martingales II (3/2/22)
Lecture 22: Motivating the Stochastic Integral (3/7/22)
Lecture 23: Constructing the Itô Integral (3/9/22)
Lecture 25:
Lecture 26: Proof of Itô's Lemma (3/28/22) (Reference: Øskendal 4.1-4.2)
Lecture 27: Martingale Representation Theorem (3/30/22) (Reference: Øskendal 4.3)
Lecture 28: Intro to Stochastic Differential Equations (4/1/22) (Reference: Øskendal 5.1)
Lecture 29: Existence and Uniqueness (4/4/22) (Reference: Øskendal 5.2)
Lecture 30: Solving Examples of SDE's (4/6/22)
Lecture 31: Diffusions and the Dynkin Formula (4/8/22) (Reference: Øskendal 7.1-7.4)
Lecture 32: Applications of Dynkin's Formula (4/11/22)
Lecture 33: Girsanov Theory (4/13/22)
Lecture 34: Interpolating Processes, Gaussian Concentration and Dimension Reduction (4/15/22)
Homework Solution Sketches:
Suggested Literature:
Markov Chains
Markov Chains and Mixing Times by Levin and Peres
A First Look at Stochastic Processes by Rosenthal
Adventures in Stochastic Processes by Resnick
Discrete Martingales
Probability with Martingales by Williams
A First Look at Rigorous Probability Theory by Rosenthal
Brownian Motion, Stochastic Calculus and Stochastic Differential Equations
Brownian Motion and Stochastic Calculus by Karatzas and Shreve
Brownian Motion by Mörters and Peres
Sotchastic Calculus and Financial Application by Steele
Stochastic Differential Equations by Øskendal