Statistics 386: Brownian Motion and Stochastic Calculus
Spring 2009

Instructor: Professor Steve Lalley
Office: 118 Eckhart Hall
Office Hours: Wednesday 1:00 - 2:00
Phone: 702-9890
E-mail: lalley@galton.uchicago.edu

This course will be a rigorous introduction to continuous-time stochastic processes, meant for students with a solid grounding in the measure-theoretic foundations of probability theory. The main topics will be :

• Brownian motion
• Levy processes and subordinators
• Ito calculus
• SDEs and diffusion processes.

Recommended Reading: R-Rated

• Stochastic Differential Equations: An Introduction with Applications (6th edition) by B. Oksendal

Recommended Reading: X-Rated

• Continuous Martingales and Brownian Motion by D. Revuz and M. Yor
• Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve

Lecture Notes:

• Discrete-Time Martingales Version: April 1 2005
• Wiener Process Version: April 1 2009 (minor correction)
• Gaussian Processes and Kolmogorov-Chentsov Theorem Version: April 7 2009
  
• Levy Processes and Subordinators Version: April 12 2007
  
• Ito Integral Version: May 12, 2009
  

Problem Sets:

• Homework 1
• Homework 2 Problem 1(E) corrected April 14
  
• Homework 3 The hint for Problem 2 has been corrected April 22
  
• Homework 4
• Homework 5 Minor corrections to Problems 2(C), 8(B), 9(B) made May 19 (thanks to Rina Foygel for finding the errors).
• Homework 6