Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

Martingales and Brownian motion (MA-368, Fall 2015)

Classes: Mon, Wed, Fri, 10:00-11:00, LH-1, Mathematics department

Description: The official name of this course is Topics in probability and stochastic processes. What will be covered are discrete time martingales and Brownian motion, and if possible also the basics of Ergodic theory. To credit the course, you must be comfortable with measure theory and probability theory (the material covered in MA 361).

Grading: There will be a final exam (50%) and two mid-term exams (total 30-50%). I can give 10-15% for scribing the class notes and 5% for solving good problems, asking good questions or for maintaining an intelligent face during lectures.

Texts and other resources: The material I cover is standard and can be found in many books, but not necessarily in one of them or in the same sequence.
  1. Rick Durrett Probability: theory and examples, Chapters 5 and 8 (excellent book that covers most of the course)
  2. David Williams Probability with martingales. Chapters 9-15. (a great book for basics of martingales)
  3. Peter Mörters and Yuval Peres Brownian motion. (a wonderful book for Brownian motion)
  4. Olav Kallenberg Foundations of modern Probability. (has everything one wants in probability, except examples)
  5. Patrick Billingsley Convergence of Probability Measures Wiley India Pvt Ltd; Second edition (2013)
  6. John Walsh Knowing the odds. (I have not read this book yet, but it has garnered high praise from experts)
  7. Paley and Wiener Fourier transforms in the complex plane. Only for the very historically minded.

Notes and homeworks/links:
  • Some notes related to Martingales (last uploaded: 28/02/2020)
  • Some notes related to Brownian motion (last uploaded: 13/03/2020)

  • Homework policy: Submit at least two reasonably non-trivial problems per week. The creditors may discuss with each other and partition the problem set among themselves or work together. If there is some question that I mention in class and you resolve it yourself, you may instead submit that as a homework problem too.

  • Homework-1
  • Homework-2
  • Homework-3
  • Homework-4
  • Homework-5