# Stochastic Processes

## Section Information

 Section Time Location Instructor Office Hours (Cupples I, Room 17) 1 MWF 1:00 PM - 2:00 PM Umrath 140 Renato Feres TTh 1:00 PM - 3:00 PM

 Please include [Math495] in the subject line of any email message that pertains to this course. This will help avoid that I accidentally delete your still unread message. My e-mail address is feres@math.wustl.edu.

## Text

Introduction to Stochastic Processes, 2nd Edition, by Gregory F. Lawler
Chpman & Hall, 2006

## Topics to be covered

This course is an introduction to stochastic processes. Topics to be covered are:
• Finite Markov chains;
• Countable Markov chains;
• Continuous time Markov chains;
• Optimal stopping;
• Martingales;
• Renewal processes and queues
• Elements of MCMC
• Brownian motion
• Stochastic integration

## Mathematics software

This is mainly a "theory" course and computer work is not as central to it as it was in Statistics, Math 3200.
Homework assignments will, nevertheless, contain a mixture of questions, some more theoretical involving proofs or computations by hand, and others involving computer work.

You may use any system for mathematics programming you wish (for example, Matlab, Mathematica, Maple, Python, etc.), but I recommend using R because this is what I will use when writing solutions to the problem sets.

In the R computing main page you'll find instructions for downloading and installing R and general documentation. In particular, the manual An Introduction to R is a useful source of information.

Although the plain R program is nice enough in my opinion, there are free programs that provide more friendly user interface environments. For example, I like RStudio.

It is very likely that most of you have used R before, but I do not assume that you have. The information provided in the problem sets and occasionally in class should be enough for the needs of the course even if you are using it for the first time.

We plan to have 9 weekly homework assignments, one midterm, and one final exam. They will all have the same weight. The exams will be take-home and will not be fundamentally different from the homework sets. The main difference is that I plan to grade the exams mostly myself (with perhaps some TA help) and will likely be more strict in grading them. The mid-term homework-exam will be due 2/20/2015 and the final homework-exam will be due 4/24/2015.

The final score will be the simple mean (with equal weights) of the top 10 scores (adjusted so that the maximal score is 100). This means that you can drop the least score of the 11.

The final score will be translated into a letter grade of A, B, C, D, F (with plus and minus shadings) in a way that is not harsher than the following table (if the grade distribution of the whole class is significantly lower than usual, then "curving" may be considered, but it is unlikely to be needed):

 Numerical Range Letter Grade [85, 100] A [70, 85) B [60, 70) C [50, 60) D [0, 50) F

In assigning pluses and minuses, I will divide the final scores in each letter roughly evenly. Please note: by this I do not mean, for example, that A- corresponds to the interval [85, 90) or that A corresponds to [90, 95). Rather, the scores in each letter range indicated in the table will be divided in roughly equal numbers so that the number of scores corresponding to A-, A, A+ will be approximately the same. Therefore, the exact grade cut-offs for the signs will not be known until all the scores are in.

## Homework

The plan is to have a total of 11 HW/exams assignments. Of these 11, the 10 best scores will count towards the total grade score.

Assignments should be turned in on the due date (Fridays) at the beginning of class.
You are allowed and encouraged to collaborate on your assignments.
Late assignments will not be accepted except under very special circumstances and never after the solutions have been posted.

## Course plan

Please keep in mind that the plan given below is somewhat tentative. Sections and problems may occasionally be changed.

 Week Textbook section covered Jan 12 - Jan 16 Homework 0 (not for grade; won't be collected) 0.1, 0.2, 0.3, 1.1 Jan 21 - Jan 23 Homework 1: 1/23 Solutions 1.1, 1.2, 1.3, 1.4 Jan 26 - Jan 30 Homework 2: 1/30 Solutions 1.5, 1.6, 2.1 Feb 02 - Feb 06 Homework 3: 2/6 Solutions 2.2, 2.3, 2.4, 3.1 Feb 9 - Feb 13 Homework 4: 2/13 Solutions 3.1, 3.2, 3.3, 3.4 Feb 16 - Feb 20 Homework 5: 2/20 Solutions 3.4, 4.1, 4.2, 4.3 Feb 23 - Feb 27 Homework 6: 1/27 Solutions 5.1, 5.2, 5.3, 5.4 Mar 02 - Mar 06 Homework 7: 3/6 Solutions 5.5, 5.6, 6.1 Mar 16 - Mar 20 6.2, 6.3, 6.4 Mar 23 - Mar 27 Homework 8: 3/27 Solutions 7.1, 7.2, 7.3 Mar 30 - Apr 03 Homework 9: 4/3 Solutions 7.4, 8.1, 8.2, 8.3 Apr 06 - Apr 10 Homework 10: 4/10 Solutions 8.4, 8.5, 8.6, 8.7, 8.8 Apr 13 - Apr 17 9.1, 9.2, 9.3, 9.4 Apr 20 - Apr 24 Homework 11: 4/24 Solutions 9.5, 9.6, 9.7, 9.8

Renato Feres
feres@math.wustl.edu
Cupples I, Room 17
(314) 935 - 6752 (office phone)

Department of Mathematics
Washington University
Campus Box 1146
Saint Louis
Missouri, 63130 USA.

Last Updated: January 10, 2015