# Stochastic Processes

## Section Information

 Section Time Location Instructor Office Hours (Cupples I, Room 17) 1 MWF 1:00 PM - 2:00 PM Mallinckrodt 305 Renato Feres Tu-Th 12:00 PM - 2:00 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 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;
• 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.

I plan to have 10 (or 9) weekly homework assignments (I'm giving myself some room for adjusting the schedule should it be needed), one midterm, and one final exam. The exams will be in class, 50 minutes long, and will be based on content covered by the homework sets. The mid-term will be on 3/04/2016 and the final on 4/22/2016. Note that there will be a final homework due 4/29/2016. The worst homework score will be dropped. Homework assignments, midterm, and final exams will count equally towards your final grade.

The final score will be the simple mean (with equal weights) of the top 9 (or 8) homework scores plus the two exams (each adjusted so that the maximal score is 100). This means that you can drop the least homework score.

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, to you as well as to me, until all the scores are in.

## Homework

The plan is to have a total of 10 (or 9, see above) HW/exams assignments. Of these, the 9 (or 8) best scores will count towards the total grade score.

Assignments should be turned in on the due date (Fridays) at the beginning of class.

Under exceptional circumstances, I may accept them at a different time on Friday, but I plan on posting solutions on Saturday. Late assignment will not be accepted after solutions are posted.

You are both allowed and encouraged to collaborate on your assignments.

## 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 20 - Jan 22 Homework 0 (not for grade; won't be collected) 0.1, 0.2, 0.3 Jan 25 - Jan 29 Homework 1: 1/29 Solutions 1.1, 1.2, 1.3, 1.4 Feb 01 - Feb 05 Homework 2: 2/05 Solutions 1.4, 1.5, 1.6 Feb 08 - Feb 12 Homework 3: 2/12 Solutions 2.1, 2.2, 2.3, 2.4 Feb 15 - Feb 19 Homework 4: 2/19 Solutions 3.1, 3.2, 3.3, 3.4 Feb 22 - Feb 26 Homework 5: 2/26 Solutions 3.4, 4.1, 4.2, 4.3 Feb 29 - Mar 04 Midterm: 3/04 Solutions 5.1, 5.2, 5.3 Mar 07 - Mar 11 Homework 6: 3/11 Solutions 5.4, 5.5, 5.6 Mar 21 - Mar 25 7.1, 7.2, 7.3 Mar 28 - Apr 01 Homework 7: 4/01 Solutions 7.3, 7.4, 8.1, 8.2 Apr 04 - Apr 08 Homework 8: 4/08 Solutions 8.3, 8.4, 8.5 Apr 11 - Apr 15 Homework 9: 4/15 Solutions 8.6, 8.7, 8.8, 9.1, 9.2 Apr 18 - Apr 22 Final: 4/22 Solutions 9.3, 9.4 Apr 25 - Apr 29 Homework 10: 4/29 Solutions 9.5, 9.6, 9.7

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 11, 2016