STAT 219 / MATH 136 – Fall 2010
Course Goals:
This course prepares students for a rigorous study of Stochastic Differential Equations, as done in Math236/Stat316. Towards this goal, we cover – at a very fast pace – elements from the material of the (Ph.D. level) Stat310/Math230 sequence, emphasizing the applications to stochastic processes, instead of detailing proofs of theorems. A critical component of Stat219/Math136 is the use of measure theory.

The Stat217-218 sequence covers many of the same ideas and concepts as Stat219 but from a different perspective. The Stat217-218 sequence can be seen as an extension of undergraduate probability (e.g. Stat116) in both level of mathematical sophistication (i.e. no measure theory). Thus, it is possible, and in fact recommended, to take both Stat217-218 and Stat219 for credit. However, be aware that Stat217-218 alone is NOT adequate preparation for Math236.

Main topics of Stat219/Math136 include: introduction to measurable, Lp and Hilbert spaces, random variables, expectation, conditional expectation, uniform integrability, modes of convergence, stationarity and sample path continuity of stochastic processes, examples such as Markov chains, branching, Gaussian and Poisson processes, martingales and basic properties of Brownian motion.

Prerequisites:
Students should be comfortable with probability at the level of Stat116/Math105/Math151 and with real analysis (a.k.a. advanced calculus) at the level of Math115. For a good review of undergraduate probability see the optional Grimmett & Stirzaker text. Appendix A of the optional Rosenthal text includes a brief review of prerequisite material from real analysis.

Required text:
The only required text for this course is Amir Dembo's lecture notes. You will be expected to read the scheduled sections in the notes before class. Below is an estimated schedule for the course.

Week beginning Monday Wednesday Friday
9/20 1.1 1.1-1.2.2 1.2.2, 1.2.3
9/27 1.3.1,1.4.1 1.3.2, 1.4.1 1.4.2, 1.4.3
10/4 2.1,2.2 2.3 2.4
10/11 3.1 3.2 3.3
10/18 5.1 Review Midterm
10/25 4.1.1, 4.1.2 4.1.3 4.2
11/1 4.3.1 4.3.1,4.3.2 4.3.2,5.2
11/8 4.4.2, 4.5 4.4.1, 5.3 4.6
11/15 6.1 6.1 6.2
11/22
11/29 6.2 6.3 Review
12/6 Final


Suggested references:
Students are encouraged to consult other references. For your convenience, a copy of each book listed below is on hold in the Math & Computer Science library.
  • Grimmett and Stirzaker, Probability and Random Processes
  • Karlin and Taylor, A First Course in Stochastic Processes
  • Lawler, Stochastic Processes
  • Lefebvre, Applied Stochastic Processes
  • Rosenthal, A First Look at Rigorous Probability Theory
  • Ross, Stochastic Processes