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 2010) and will be followed by (and required for) Math 275C (Spring 2011) 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: MF 2:50-3:50 in MS 6228)**Time and place:**Lectures MWF 2 in MS 6221**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.

**[Dec 15]:**There will be no discussion section this quarter.

**Lec 1 [Jan 3]:**Syllabus. Conditional expectation I: definition, existence, uniqueness. Sec 5.1.**Lec 2 [Jan 5]:**Conditional expectation II: examples, properties, (regular conditional probabilities). Sec 4.1.**Lec 3 [Jan 7]:**Martingales I: definition, examples. Sec 5.2.**Lec 4 [Jan 10]:**Martingales II: stopping times, betting systems. Sec 5.2.**Lec 5 [Jan 12]:**Martingale convergence theorem. Sec 5.2.**Lec 6 [Jan 14]:**Branching processes. Sec 5.3.**No lecture on Jan 17 (Martin Luther King, Jr, holiday).****Lec 7 [Jan 19]:**Martingales in L2. Back to branching processes. Sec 5.4.**Lec 8 [Jan 21]:**Martingales in L2 (continued). Sec 5.4.**Lec 9 [Jan 24]:**Martingales in Lp. Sec 5.4.**Lec 10 [Jan 26]:**Uniform Integrability. Sec 5.5.**Lec 11 [Jan 28]:**Levy's Upward Theorem. Sec 5.5.**Lec 12 [Jan 31]:**Levy's Downward Theorem. Sec 5.6.**Lec 13 [Feb 2]:**Optional Sampling Theorem. Sec 5.7.**Lec 14 [Feb 4]:**Stationary Processes. Sec 7.1.**Lec 15 [Feb 7]:**Birkhoff's Ergodic Theorem I. Sec 7.2.**Lec 16 [Feb 9]:**Birkhoff's Ergodic Theorem II. Sec 7.2.**Lec 17 [Feb 11]:**Subadditive Ergodic Theorem. Sec 7.4.**Lec 18 [Feb 14]:**Review: multivariate Gaussian distribution. Sec 3.9. (Sec 1.2 in [L].)**Lec 19 [Feb 16]:**Definition and construction of Brownian motion. Sec 8.1. (Sec 1.4, 1.5 in [L].)**Lec 20 [Feb 18]:**Path properties I. Sec 8.1, 8.4. (Sec 1.6 in [L].)**No lecture on Feb 21 (President's Day holiday).****Lec 21 [Feb 23]:**Path properties II. Sec 8.1, 8.4. (Sec 1.6 in [L].)**Lec 22 [Feb 25]:**Path properties III. Sec 8.1, 8.4. (Sec 1.6 in [L].)**Lec 23 [Feb 28]:**Markov property. Sec 8.2. (Sec 1.7 in [L].)**Lec 24 [Mar 2]:**Strong Markov property I. Sec 8.3. (Sec 1.8 in [L].)**Lec 25 [Mar 4]:**Strong Markov property II. Sec 8.3. (Sec 1.8 in [L].)**Lec 26 [Mar 7]:**Martingale property. Sec 8.5. (Sec 1.9 in [L].)**Lec 27 [Mar 9]:**Donsker's invariance principle I. Sec 8.6. (Sec 1.10, 1.11 in [L].)**Lec 28 [Mar 11]:**Donsker's invariance principle II. Sec 8.6. (Sec 1.10, 1.11 in [L].)

**Hwk 1 [Due in class Jan 10 (MONDAY)]:**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 Jan 24 (MONDAY)]:**Choose 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 7 (MONDAY)]:**Choose 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 18 (FRIDAY)]:**Choose 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 4 (FRIDAY)]:**Choose 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 11 (FRIDAY)]:**Choose 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: Feb 7, 2011.