Math 275 is an introduction to *rigorous probability* at the graduate level.
The Winter quarter will give an introduction to stochastic processes
in both discrete and continuous time, including:
martingales;
stationary processes;
and brownian motion.

While you may have encountered some of these topics in an undergraduate probability course, we will take a much deeper look at them here. This course follows (and requires the equivalent of) Math 275A (Fall 2011) and will be followed by (and required for) Math 275C (Spring 2012) which will develop further the theory of stochastic processes in continuous time with an emphasis on Markov processes. It should appeal both to students interested in pure mathematics (esp. analysis) and in applications (esp. physics, engineering, biology, economics).

*Prerequisites:*
Math 275A or equivalent.

**Instructor:**Sebastien Roch (Office hours: M 11-11:50, F 2-2:50 in MS 6156)**Time and place:**Lectures MWF 10 in MS 5233**Required Text:**- [D] Probability: Theory and Examples (4th Edition) by Durrett (available online)

**Optional Texts.**I will also occasionally refer to the following excellent texts:- [L] Continuous Time Markov Processes: An Introduction by Liggett (for Brownian motion part)
- [MP] Brownian Motion by Morters and Peres (for Brownian motion part)
- [W] Probability with Martingales by Williams (for martingales part)

**Grades**will be based on homework to be assigned mostly from Durrett's book. There will be 6 assignments. All homework assignments will contribute equally to the final grade. The lowest homework score will be dropped.

**[Jan 8]:**There will be no discussion section this quarter.**[Jan 13]:**Due date for homework 1 was pushed back to Friday, Jan 20.

UPDATE [Sep 1, 2015]: The latest version of the lecture notes can be accessed here.

The following lecture notes are strongly influenced by the references above. They have not been proofread very carefully, so use them at your own risk. A file containing all the notes is here.

**Lec 1 [Jan 9]:**Conditional expectation I: definition, existence, uniqueness. Sec 5.1.**Lec 2 [Jan 11]:**Conditional expectation II: examples, properties, (regular conditional probabilities). Sec 5.1.**Lec 3 [Jan 13]:**Martingales I: definition, examples. Sec 5.2.**No lecture on Jan 16 (Martin Luther King, Jr, holiday).****Lec 4 [Jan 18]:**Martingales II: stopping times, betting systems. Secs 4.1, 5.2.**Lec 5 [Jan 20]:**Martingale convergence theorem. Sec 5.2.**Lec 6 [Jan 23]:**Branching processes. Sec 5.3.**Lec 7 [Jan 25]:**Martingales in L2. Back to branching processes. Sec 5.4.**Lec 8 [Jan 27]:**Martingales in Lp. Sec 5.4.**Lec 9 [Jan 30]:**Read Martingales in L2 (continued). Sec 5.4.**Lec 10 [Feb 1]:**Uniform Integrability. Sec 5.5.**Lec 11 [Feb 3]:**UI Martingales. Sec 5.5, 5.6.**Lec 12 [Feb 6]:**Optional Sampling Theorem. Sec 5.7.**Lec 13 [Feb 8]:**Stationary Processes. Sec 7.1.**Lec 14 [Feb 10]:**Birkhoff's Ergodic Theorem I. Sec 7.2.**Lec 15 [Feb 13]:**Birkhoff's Ergodic Theorem II. Sec 7.2.**Lec 16 [Feb 15]:**Subadditive Ergodic Theorem. Sec 7.4.**Lec 17 [Feb 17]:**Review: multivariate Gaussian distribution. Sec 3.9. (Sec 1.2 in [L].)**No lecture on Feb 20 (President's Day holiday).****Lec 18 [Feb 22]:**Definition and construction of Brownian motion. Sec 8.1. (Sec 1.4, 1.5 in [L].)**Lec 19 [Feb 24]:**Path properties I. Sec 8.1, 8.4. (Sec 1.6 in [L].)**Lec 20 [Feb 27]:**Path properties II. Sec 8.1, 8.4. (Sec 1.6 in [L].)**Lec 21 [Feb 29]:**Markov property. Sec 8.2. (Sec 1.7 in [L].)**Lec 22 [Mar 2]:**Strong Markov property. Sec 8.3. (Sec 1.8 in [L].)**Lec 23 [Mar 5]:**Martingale property. Sec 8.5. (Sec 1.9 in [L].)**Lec 24 [Mar 7]:**Skorokhod embedding. Sec 8.6. (Sec 1.10, 1.11 in [L].)**Lec 25 [Mar 9]:**Donsker's invariance principle. Read Sec 8.6. (Sec 1.10, 1.11 in [L].)**Lec 26 [Mar 12]:**Applications of Donsker's invariance principle I. Read Sec 8.6. (Sec 1.10, 1.11 in [L].)**Lec 27 [Mar 14]:**Arcsine laws for BM and RWs. Read Sec 5.4 in [MP].**Lec 28 [Mar 16]:**Random walks: Recurrence. Sec 4.2.

**Hwk 1 [Due in class Jan 20 (FRIDAY)]:**Read Section 4.1. Submit FIVE among exercises 4.1.3, 4.1.6, 4.1.8, 5.1.1, 5.1.5, 5.1.9, 5.1.13.**Hwk 2 [Due in class Feb 3 (FRIDAY)]:**Submit FIVE among exercises 5.2.6, 5.2.9, 5.3.12, 5.4.4, 5.4.5, 5.4.6, 5.4.7.**Hwk 3 [Due in class Feb 17 (FRIDAY)]:**Submit FIVE among exercises 5.5.1, 5.5.5, 5.5.6, 5.5.8, 5.6.5, 5.7.3, 5.7.9.**Hwk 4 [Due in class Feb 24 (FRIDAY)]:**Submit FIVE among exercises 7.1.1, 7.2.1, 7.2.2, 7.4.1 (which refers to Example 7.4.3), 7.5.1, 7.5.2, 7.5.3.**Hwk 5 [Due in class Mar 9 (FRIDAY)]:**Submit FIVE among exercises 8.1.1, 8.2.1, 8.2.2, 8.2.3, 8.3.1, 8.3.4, 8.3.7.**Hwk 6 [Due in class Mar 16 (FRIDAY)]:**Submit FIVE among exercises 8.4.1, 8.4.2, 8.5.1, 8.5.2, 8.5.4, 8.5.5, 8.5.6.

Last updated: March 20, 2012.