Math 495
Stochastic Processes

Professor Wickerhauser




  • PolyaUrn.txt, an R function (contained in a plain text file) to compute the red fraction in Polya's Urn after a specified number of balls are added to initial red and green ball contents, as well as R commands to perform two numerical experiments and display the results.
  • OptimalStop.txt, an R function (contained in a plain text file) to compute the state valuations for the Markov chain of the dice game example in Chapter 4.
  • RateMatrices.txt, some wxMaxima code (contained in a plain text file) to exponentiate the infinitesimal generator matrices of two examples in Chapter 3 of the textbook.
  • Walk1d.txt, an R function (contained in a plain text file) to plot a random walk in 1 dimension with specified length and transition probabilities.


Topics. Content varies with each offering of the course. Past offerings have included such topics as random walks, Markov chains, Gaussian processes, empirical processes, Markov jump processes, and a short introduction to martingales, Brownian motion and stochastic integrals.

Prerequisites. Math 318 (Calculus of Several Variables) and Math 493 (Probability), or permission of instructor.

Time. Classes meet Mondays and Wednesdays, 4:00 pm to 5:30 pm, in Duncker Hall room 101.

Text. The lectures will follow Gregory F. Lawler's textbook Introduction to Stochastic Processes, second edition, ISBN 978-1-58488-651-8, Chapman and Hall/CRC, 2006.

Homework. You are encouraged to collaborate on homework and to work additional exercises from the indicated problem sections, although the homework grade will be based only on the exercises listed below. Please return your solutions to the instructor by the end of class. Problem sets will be assigned as follows:
Solutions are due at the end of class on the due date. Late homework will not be accepted. The problems will often require a complete proof. The homework will be judged for correctness and clarity. If you use a computer for solution, please include your well-documented computer code which will be judged in part for its understandability. Please print and submit: (i) the code, with a comment for every line, (ii) the input you gave it, and (iii) the output it produced, for at least one example run.

Tests. There will be one midterm examination in class on Wednesday, March 7th, 2018. No reference material or electronic devices will be allowed.
There will be one cumulative take-home final examination emphasizing the remaining material, due on Friday, May 4th, 2018 by 8:00 pm in my office, room 105a, Cupples I Hall.

Grading. One score will be assigned for homework, one for the midterm examination, and one for the final examination. These three will contribute in respective shares of HW 50%, MT 20%, and FE 30% to the course score. Letter grades, computed from the course score class average and standard deviation, will be at least the following:

Course score at least:90%80%70%60%
Letter grade at least:ABCD

Students taking the Cr/NCr or P/F options will need a grade of D or better to pass.

Computing. Students are encouraged to use computers for both symbolic and numerical computations.

Office Hours. See me in Cupples I, room 105a, Mondays and Wednesdays after class, or by appointment.

Questions? Return to M. Victor Wickerhauser's home page for contact information.