Math 5040: Stochastic Processes and Simulation I, Fall 2006



Time and place: 9:40--10:30 MWF in LS 111.

Instructor: S. Ethier (Prof.), JWB 119, 581-6148, ethier@math.utah.edu. Office hours are 2:00--2:50 MWF unless announced otherwise. Other times are available by appointment.

Text: Essentials of Stochastic Processes by Rick Durrett (Springer, 1999). Available at amazon for $72.37. List of typos.

Prerequisite: Math 5010. This is absolutely essential. Also, some knowledge of computer programming will be helpful.

Topics covered: We will cover the whole book over the course of two semesters (Markov chains, martingales, Poisson processes, continuous-time Markov chains, renewal theory, and Brownian motion). This will probably require supplementing the material in the book. But we will begin with some lectures on simulation, which the book does not cover. For that material a good reference is Sheldon Ross's Introduction to Probability Models, Chapter 11. This book will be on reserve in the Math Library, and I will post lecture notes on this web page, at least for the simulation material. Those with an interest in biology may want to consider Stochastic Processes in Physiology instead. It is a 6000-level course but is only one semester and has slightly different prerequisites.

Assignments: There will be textbook assignments and simulation assignments. For the simulation assignments, you can use whatever programming language you prefer (C, FORTRAN, Pascal, BASIC, Excel, Mathematica, MatLab, Maple, Java, etc.). You will turn in your program (with documentation, i.e., comments explaining what your instructions mean) and output on paper. I do not want an electronic file. The purpose of the assignments is to learn the material. Therefore, working in groups is not allowed because the result is usually that one person does the work and the others copy it. Evidence of collaboration or plagiarism will result in severe penalties for all involved.

Grades: There will be a midterm exam on about October 16. There will be a final exam on Monday, December 11 at 8 a.m. Grades will be based on homework (30%), simulation assignments (20%), midterm exam (20%), and final exam (30%).

Lectures on simulation: Lecture 1, 8/23. Lecture 2, 8/25. Lecture 3, 8/28. Lecture 4, 8/30. Lecture 5, 9/01. Lecture 6, 9/06. Lecture 7, 9/08. Lectures 1-7 in a single file.

Problem Set 1. Due 9/27. Solutions.

Lectures on Markov chains: Lecture 8, 9/11. Lecture 9, 9/13. Lecture 10, 9/15. Lecture 11, 9/18. Lecture 12, 9/20. Lecture 13, 9/22. Lecture 14, 9/25. Lecture 15, 9/27. Lecture 16, 9/29. Lecture 17, 10/2. Lecture 18, 10/4. Lecture 19, 10/9. Lecture 20, 10/11. Lecture 21, 10/18. Lecture 22, 10/20. Lecture 23, 10/23. Lecture 24, 10/25. Lecture 25, 10/27. Lecture 26a, 10/30. Lectures 8-26a in one file.

Problem Set 2. Durrett, page 88. 1, 5, 6, 8, 14, 17, 20, 25, 29, 37. Due 10/11. Solutions.

Topics for the midterm exam on Monday Oct. 16. Old midterm. We went over this in class (except problem 4, which we haven't covered). This year's midterm, with solutions.

Lectures on Markov chain Monte Carlo: Lecture 26b, 10/30. Lecture 27, 11/01. Lecture 28, 11/03.

Problem Set 3. Durrett, page 96. 39, 45--49, 51, 53. In addition, 3 simulation problems. Due 11/13. Solutions.

Lectures on conditional expectation and martingales: Lecture 29, 11/06. Lecture 30, 11/08. Lecture 31, 11/10. Lecture 32, 11/13. Lecture 33, 11/15. Lecture 34, 11/17. Lecture 35, 11/20. Lecture 36, 11/22. Lecture 37, 11/27. Lecture 38, 11/29. Lecture 39, 12/04. Lectures 29-39 in one file.

Problem Set 4. Durrett, page 121. 2, 5, 6, 7, 14, 17, 19, 20. Due Dec. 6. Some hints may be given in class. Solutions.