Statistics 385: Brownian Motion and Stochastic Calculus
Fall 2016

Instructor: Professor Steve Lalley
Office: 118 Eckhart Hall
Office Hour: Thursday 1:00 - 2:00
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 (PG)
• Brownian Motion and Stochastic Calculus by Ioannis Karatzas and Steve Shreve (R)
• Foundations of Modern Probability by Olav Kallenberg (R)
• Continuous Martingale and Brownian Motion by Revuz and Yor (X)

Background Reading:

• Independence
• Conditional Expectation
• Discrete-Time Martingales

Lecture Notes:

• Brownian Motion
• Harmonic Functions and Brownian Motion in Several Dimensions
• Levy Processes revised October 23 2017
• Continuous Martingales I. Fundamentals
• The Ito Integral (revised November 14 2016)
• Stochastic Differential Equaitions revised December 2, 2016

Assignments:

• Assignment 1 due October 10
• Assignment 2 due October 24
• Assignment 3 due November 7
• Assignment 4 due November 23
• Assignment 5 due December 7 (corrected November 30 2016)