**Mathematics
495**

Spring,
2013

**Instructor: **Edward N. Wilson

Cupples I, Room 18

E-mail: enwilson@math.wustl.edu

Office Tel: 935-6729 (O.K. to leave messages, better to use e-mail)

**Office Hours: ** MW 2-3,
4-5 and by appointment

**Class Meeting Times
and Place: **MWF 3-4 Lopata 229

**
**

**Textbooks: 1. Introduction To Stochastic Processes,
**(paperback edition), by Paul G. Hoel, Sidney C. Port, Charles J. Stone, published in 1972 by
Waveland/Houghton-Mifflin.

**2.
Introduction to Stochastic Processes, 2 ^{nd} Edition, **by
Gregory F. Lawler, published in 2006 by Chapman&Hall/CRC.

**Overview: **We'll use both of the textbooks,
trying to get through well over half of each one during the course of the
semester. Homework assignments
will involve problems from each book.
We'll refer to these books as HPS (Hoel, Port, & Stone) and L
(Lawler). Although there's a
considerable amount of overlap between HPS and L, HPS is mathematically more
precise and less "user-friendly" while L only tries to "give the main ideas
without proofs" and is "less intimidating"; sadly, L is also very sloppy with
notations and often very hazy with "explanations". Especially in the early stages, students will probably find L
easier to read since Chapters 1 and 2 of HPS are hard to read. On the other hand, Chapter 3 in HPS
is much better written than Chapter 3 in L and in Chapters 8 and 9, L "jumps off
the high board" into modern stochastic calculus without doing any of the
traditional stochastic calculus presented nicely by HPS in Chapters 4-6. Since HPS is regarded as a classic book
in stochastic processes, having it in one's personal library is beneficial to
everyone who uses stochastic processes.
Those who want to use the very fancy Ito calculus sketched in Chapter 9
of L will need to acquire later on a book giving a more sophisticated treatment
of this subject than that given by L.

We'll begin with the treatment of finite state space Markov chains. These are covered in Chapter 1 of L with Chapter 2 of L moving on to infinite state space Markov chains. On the other hand, HPS do the finite and infinite state space cases simultaneously in their Chapters 1 and 2. We'll then move on to continuous time jump processes (Chapter 3 in each of HPS and L) and martingales (Chapter 5 in L) As time permits, we'll skim through Chapters 4-6 of HPS, Chapter 8 of L on Brownian motion, and a light treatment of the Ito calculus following Chapter 9 of L.

In brief, stochastic processes are the probabilistic version of dynamical systems.

Both of these areas are enjoying great popularity with those who want to develop a model for how some complicated phenomenon (economic development, business cycles, the weather, ecological systems, ... ) evolves over time. In dynamical systems, deterministic models are used with a fixed transition function assumed to exist which, if we only knew a formula for it, would precisely predict the next stage of evolution based on the current stage (or state) . Stochastic processes also want to predict futures based on the current state but do so only in probabilistic fashion, giving probabilities for entering various states tomorrow based on today's state. Linear algebra plays a very large role in both subjects. For the finite state space Markov chains with discrete time increments (days, weeks,...) , this comes down to studying certain types of matrices called Markov matrices. For the infinite state space Markov chains with discrete time increments, we have to go on to Markov matrices with infinitely many rows and columns. Linear algebra is still important but we have to rely more on probability to prove the basic theorems. Not surprisingly, the results aren't as "clear cut" as in the finite state space case. When we move to the more realistic setting of continuous time Markov processes, we're essentially obliged to limit attention to random variables having nice probability density functions and to use calculus tools. Basic calculus is still the main tool for the material in Chapters 4-6 of HPS. However, to understand the Ito formulas (modern stochastic calculus), one has to rely on measure theory tools. To put it mildly, this makes modern stochastic calculus a very difficult subject.

**Homework: **There will be weekly homework
assignments to be written up and handed in during a designated class. During the week of an exam, we'll
suspend homework.

The homework problems will be drawn from both HPS and L. Each has a wide range of problems, some considerably more difficult than others. When a number of students have trouble with a particular problem, we'll try to go over it in class. Homework will be an important part of the course for the usual reasons: it's virtually impossible to learn a mathematical subject without working through problems.

**Examinations: **There will be an in-class exam
either on Monday, February 12 or, if a majority of the class prefers it,
Friday, Feb. 9. In late March or
early April, there will be a take-home exam. Since the instructor will be out of town during final exam
week, we'll have to give an early final exam: either 1 hour on Friday, Apr. 26 (last scheduled class day)
or, preferably, two hours on Saturday, April 27. Guidance on the style of each exam and the material covered
by the exam will be given in class.

In the absence of documentation from the Health Service or the Dean's Office attesting to valid reasons for a student to miss an exam, a grade of 0 will be recorded

for the missed exam. Otherwise, to the extent possible, make-up exams will be avoided by simply excusing the student from the missed exam and basing the course grade on remaining exam scores and homework.

**Academic Integrity: ** When there is evidence strongly
suggesting that cheating took place on an exam, the evidence will be forwarded
to the Arts and Sciences Integrity Committtee. If, after a hearing with the instructor and affected
student(s), a majority of the Committee members are convinced that cheating did
occur, the Committee will assess a penalty and inform the instructor and
sudent(s) of its decision.
For exam cheating, the most common penalties are either a failing grade
on the exam or a failing grade in the course. In brief, *please don't cheat on
exams* since cheating cases, however they turn out, are very painful for all
concerned.

In
sharp contrast to the above paragraph, students are *encouraged *to discuss homework problems with other students in the
course and are *welcome * to use hints provided by the instructor
during office hours. But each student must write up his/her own solutions and,
rather than simply copying someone else's solution, think through the problem
while writing it up and use his/her own notations. When two or more students turn in essentially identical
solutions and it's deemed to be virtually certain that this couldn't have
happened accidentally, the maximum score for the assignment will be divided
among the collaborators. Thus, if
3 students turn in identical solutions on a homework set worth a maximum of 12
points, 4 will be the maximum for each collaborator.

**Grading Scale: ** The first exam and the take-home exam will each account for 20%
of the course grade average, homework for 30%, and the final exam will account for
the remaining 30%. Letter grades
for exams and homework/course averages will be determined as follows:

A 90-100%

B 80-89%

C 65-79%

F below 65%

In no case will final grades be lower than indicated by this scaling. However, the instructor reserves the right to switch to a more lenient scaling at the end of the semester if he decides that it is warranted. Also, it's quite likely the final exam will be optional for those who have a B or better pre-final average and elect not to try to improve this grade by taking the final exam.