Statistics 385: Brownian Motion and Stochastic Calculus
Fall 2014

Instructor: Professor Steve Lalley
Office: 118 Eckhart Hall
Office Hour: Thursday 2:00 - 3:00
Phone: 702-9890
E-mail: lalley@galton.uchicago.edu

This course is the third in a three-quarter sequence in measure-theoretic probability. It is meant for students with a solid grounding in real analysis, including measure and integration, at the level of Stat 381 or Math 312, and some knowledge of conditional expectation and discrete-time martingale theory. The first part of the course will deal with Brownian motion and several related processes. The bulk of the course will be devoted to the basics of the Ito calculus and its use in the study of stochastic differential equations.

There will be no exams, but about 5 assignments.

Recommended Reading:

• Stochastic Differential Equations: An Introduction with Applications by Bernt Oksendal
• Brownian Motion and Stochastic Calculus by Ioannis Karatzas and Steve Shreve

Background Reading:

• Independence
• Conditional Expectation
• Discrete-Time Martingales

Lecture Notes:

• Brownian Motion
• Gaussian Processes, Wiener Isometry, and Kolmogorov-Chentsov Theorem
• Harmonic Functions and Brownian motion updated October 20, 2015
• Continuous Martingales and Local Martingales updated November 16
• Stochastic Calculus: Consequences of the Ito Formula updated November 25
• Stochastic Differential Equations

Assignments:

• Assignment 1 due October 13
• Assignment 2 due October 27
• Assignment 3 due November 10
• Assignment 4 due December 1
• Assignment 5 due December 8