Math 495 - Stochastic Processes

Spring 2005

Note: You can see me outside the set office hours, but contact me in advance to be sure I'm in.

Text: Introduction to Stochastic Processes, by Hoel, Port and Stone.

Homework and grades: The course work will consist of approximately 6 homework assignments, handed out (roughly) bi-weekly. Grades will be calculated based on your homework scores. The exact grade scale will not be decided till the end of the course. However, the final letter grade is guaranteed to be no harsher than the following:

Topics: We plan to cover the following main topics.

• Introduction to probability spaces
• Markov chains (discrete and continuous time)
• Second order processes, the Wiener process and white noise
• Introduction to stochastic differential equations
• Applications.

Homework assignments and the due dates will be posted here. Problems will mostly be taken from the textbook.

• H.W. 1 (Due on Friday, 2/04/05.) Problems: chapter 1, #10, #13, #14, #20.
• H.W. 2 (Due on Friday, 2/18/05.) Problems: chapter 1, #25, #26, #27, #33, #35, #36.
• H.W. 3 (Due on Friday, 3/04/05.) Problems: chapter 2, #3, #5, #10, #13, #17.
• H.W. 4 (Due on Friday, 3/25/05.) Problems: chapter 3, #10, #12, #16, #21, #22.
• H.W. 5 (Due on Friday, 4/08/05.) Problems: chapter 4, #4, #8, #18, #19, #20.
• H.W. 6 (Due on Friday, 4/18/05.) Problems: chapter 5, #13, #15, #16.

Lecture notes: I will occasionally write notes to supplement the textbook. A pdf file will be posted here. I will let you know in class when there is an update that makes it worth reloading. Here is another set of notes I wrote for a more advanced course, which contains some useful material: Lecture notes for Math 545 (geometry and probability).

(Potentially) useful links: You might find some of these links useful (or not). They contain lecture notes and texts for stochastic processes and probability theory that can be freely downloaded, historical notes and other resources. I'll add to this list as I find other interesting Web pages. In any event, I will not rely on any of this material for the course. If you find other useful pages, let me know so that I can add to this list.

• Stochastic process - Wikipedia.

• Stochastic Process - PlanetMath (slow).

• A summary of probabilities .

• Virtual Laboratories in Probability and Statistics.

• Topics in Probability by Wlodzimierz Bryc .

• Markov chains - Wikipedia.

• Markov chains - Mathworld.

• Random walks and electrical networks - by Doyle and Snell (great book; free download pdf file) .

• Reversible Markov Chains and Random Walks on Graphs (by Aldous and Fill: monograph in preparation).

• A Random Look at Brownian Motion.

• Course on stochastic differential equations - by Klaus Bichteler .

• Random Walks and Diffusion - lecture notes by M. Bazant .

• Stochastic Differential Equations - by Peter Zimmer .

• Signal processing .

• Hidden Markov Models .

• Bookmarks for Markov Chains (Hidden and Normal).

• Stochastic Processes with Focus in Petroleum Applications.

• Probability and Music .

• Digital music programming .

• Probability by Surprise.

• Probability Distributions.

• Java + Probability.

• Einstein's Explanation of Brownian Motion .

• Markov System Simulation.

• U of T Day Statistics Applets .

• Mathematics in Finance Applets.

• Matlab codes I.

• Matlab codes II.

• Matlab program files for Stochastic Differential Equations .

• Finite state Markov chain .

• Information Theory, Inference and Learning Algorithms - by D. MacKay (Great book; free download pdf file.)

• Andrei Andreyevich Markov.

• Simeon Denis Poisson .

• Johann Carl Friedrich Gauss.

• Norbert Wiener .

• Einstein and the history of Brownian motion .