MATHEMATICAL PROBABILITY

MATH 6383 - SPRING 2008

Instructor:

Tolya Puhalskii

Office:

CU Building, Room 613

Phone:

303-556-4811

Email:

anatolii.puhalskii@cudenver.edu

Home page:

http://www-math.cudenver.edu/$\,\tilde{}\,$puhalski/

Course web page:

http://www-math.cudenver.edu/$\,\tilde{}\,$puhalski/teaching/08-math-prob

Class hours:

Monday & Wednesday 5:30 p.m. - 6:45 p.m., CU 656

Office hours:

Monday & Wednesday 4:00 p.m. - 5:00 p.m. (or by appointment).

Purpose

Catalogue's course description.

Measurable spaces, probability measures, random variables, conditional expectations and martingales. Convergence in probability, almost sure convergence, convergence in distribution, limit theorems (law of large numbers, central limit theorem, law of iterated logarithm).

Instructor's course description.

In the preface to his classical text on probability (`Probability', Addison-Wesley, 1968), Leo Breiman speaks of the right and left hands of probability. To quote him: ``On the right is the rigorous foundational work using the tools of measure theory. The left hand `thinks probabilistically', reduces problems to gambling situations, coin-tossing, motions of a physical particle.'' While ``left-handed mentality'' often suffices for elementary applications and is useful for gaining insights, it lacks rigor, so its conclusions have to be taken with a grain of salt. More advanced applications such as mathematical finance require more sophisticated tools. Here, intuition is not as helpful and one has to start by putting probability on a solid foundation. In this ``right-handed'' course, we develop probability ``from scratch'' as a rigorous mathematical theory. We will introduce the concepts of a probability space and a probability measure, define random variables and study their properties. Limit theorems is a classical topic and continue to play an important role both in theory and applications, so we discuss them at quite some length. The course culminates with the study of martingales which are indispensable in modern probability theory. On having completed this course you will be well equipped for a study of more complicated concepts and entities such as the Brownian motion and Ito equations which feature in many models used for real-life applications.

Course objectives.

By the end of the course you should have the knowledge of the basic concepts of probability theory and the ability to apply it to do elementary proofs.

Requirements

Required text.

J. Jacod and Ph. Protter, Probability Essentials, Springer, 2000.

Prerequisites.

There is no formal prerequisite for this course. However, certain mathematical maturity is required. Also things will go more smoothly if you have been exposed to `left-handed' probability, e.g., in MATH 4810/5310, and have some analysis background as provided in MATH 4320, although no knowledge of the subject matter of these courses is assumed.

Course conduct.

I will lecture most of the time. However, due to time constraints some of the proofs will not be discussed in class. It is your responsibility to read and understand the omitted material.

Grading.

Your grade will be based on homework (30%), a take-home midterm exam (30%), and a take-home final exam (40%). The tentative exam papers due dates are Monday, March 17 and Monday, May 12. Homework will be assigned and collected regularly. You will be able to find the assignments and the dates when they are due on the course web page. The course grade will use the standard grading scale: 90%=A, 80%=B, 70%=C, 60%=D.

Course policies.

Late work is accepted at a 10% penalty. If there is a reason beyond your control that prevents you from turning in an assignment on time, you shoud tell me that beforehand.

Communication.

In addition to announcements made and handouts distributed in class, I may need to contact you between classes, which I'll do through individual and group email messages. You are expected to maintain an email address, check it regularly for messages, be sure it is working, and let me know if you change your email address. You are responsible for any messages, including assignments and schedule changes, I send you via email. You also may contact me via email, in addition to seeing me during office hours or calling me. If you'd like to speak to me outside office hours, you're encouraged to contact me in advance to set up an appointment.

Incomplete grades.

Incomplete grades (IW or IF) are not granted for low academic performance. To be eligible for an Incomplete grade, students must (1) successfully complete 75 percent of the course, (2) have special circumstances (verification may be required) that preclude them from attending class and completing graded assignments, and (3) make arrangements to complete missing assignments with the original instructor. A CLAS Course Completion agreement is strongly suggested.

Civility and academic honesty.

Adherence to the Student Code of Conduct is expected, see
http://thunder1.cudenver.edu/studentlife/studentlife/discipline.html

Students with disabilities.

The University of Colorado Denver is committed to providing reasonable accommodation and access to programs and services to persons with disabilities. Students with disabilities who need academic accommodations must register with Disability Resources and Services (DRS), 177 Arts Building, 303-556-3450, TTY 303-556-4766, FAX 303-556-2074. I will be happy to provide approved accommodations, once you provide me with a copy of DRS’s letter.

Drop deadlines.

The last day to drop this class without the drop charge is January 28, 2008. The last day to drop with a tuition refund and no transcript notation is February 6, 2008 by 5 p.m. The drop charge applies. This is an absolute deadline. The last day to drop or withdraw without a petition and a special approval from the student's dean is April 7, 2008 by 5 p.m. This is an absolute deadline.