Download American University of Kuwait

Eastern Michigan University
Math 479: Introduction to Stochastic Calculus
Spring 2011
Student Syllabus
Instructor: Dr. Ovidiu Calin
Office #: Pray-Harrold, room 516 F
email: [email protected]
________________________
Class Hours: Monday, Wednesday 5:30 pm – 8:10 pm in Alexander 218 A
________________________
Office Hours: Monday, Wednesday: 5:00 – 5:30 pm in Alexander 218 A
8:10 – 8:30 pm in Alexander 218 A
Course description
This is a gentle introduction to Stochastic Calculus. All differentiation and integration techniques
covered in Math 120 and Math 121 in the deterministic case will be covered here for the
stochastic case. Many differentiation rules, such as the product rule or chain rule have an analog
in the stochastic environment.
The course starts with basic notions of probability spaces and random variables and their
convergences, and their conditional expectations. Then it deals with a presentation of basic
stochastic processes, such as the Brownian motion, Bessel process, Poisson process and their
main properties. The rest of the material deals with the Ito stochastic integral and several
techniques of integration. The exposition is elementary, suited to the advanced undergraduate.
Course Webpage
You may check for a detailed table of contents at the page
http://people.emich.edu/ocalin/Teaching_479.htm
Prerequisites
Students are expected to have taken previously the Calculus sequence, Math 370. Also Math 325
preferred.
Textbook
The material for this course will be posted in pdf format on the course webpage.
http://people.emich.edu/ocalin/Teaching_479.htm
Technology
This course does not require any technology. However, the student might be interested to take
other related courses regarding simulations of stochastic differential equations, which do require
a great deal of programming.
Evaluation
There are two midterm exams and a final examination.
Weekly homework assigned from problems given at the end of each chapter.
The cut-offs for the final grade in this course will be calculated as follows:
A
95 %
A90 %
B+
85 %
B
80 %
B75 %
C+
70 %
C
65 %
D
60 %
E
<60 %
Attendance
Even if class attendance is not required, only few students are able to understand the material by
themselves, just reading the material. So absences will most probably affect your grade in a
negative manner. It is your best interest to show up.
Instructor Information
I am a full-time faculty member at Eastern Michigan University in the Department of
Mathematics and your education is my primary responsibility. I am happy to meet with you on
any issue regarding class activity and to provide help when you do not understand something we
have covered. You may see me in my office before the class or after class if time permits. Also,
my email address is [email protected].
More information about my teaching and research interests can be found at my webpage at
http://people.emich.edu/ocalin/
Classroom Etiquette
Please silence your cellular phones before class and, under no circumstances may you use a
phone for any purposes during class, including text messaging. If you need to make an
emergency call, please do it after leaving the class quietly.
Disabilities
Any students needed to arrange a reasonable accommodation for a documented disability should
contact Access Service Office in 203 King Hall, 487-2470.
Miscellaneous
1.
2.
3.
In the event that you must miss an exam or a quiz due date for a VALID
(emergency) AND verifiable (PRESENT PROOF) I must be notified
preferably in advance by a phone call or an email message.
Examples of unverifiable (or inadequate) reasons: oversleeping, going on a
pleasure trip, or just not feeling like coming to class.
Weather-related problems that I can verify through the news media are always
4.
5.
6.
7.
8.
valid. Do not risk your life to come to class! If school is in session I will most
likely be here, because I don't live far away. If you commute and do not feel it
is safe to travel, do not come. Give me a call and you will not be penalized for
missing the class.
Class attendance is essential. If you must miss a class, it is your responsibility
to get any handouts from the professor, notes from a classmate, and complete
the assignments. Also, the professor is available during the office hours.
Tutorial assistance is available free of charge in the Math Student Help Center
(220 Pray-Harrold). Assistance is also available through The Learning center
on campus (Bruce T. Halle Library).
If you cannot attend the official office hours, I also offer office hours by
appointment.
Every student will be assigned a number in the first day of class, called lucky
number. Please write this number together with your name on all homework
assignments and tests for the rest of the semester.
When you prepare your homework please write eligibly. The grade is given not
only for correctness but also for the presentation and style.
During exams students often do mistakes and they need to erase some parts of
the solution. It is suggested to write the exam in a black pencil and have an
eraser on hand.
Topics Covered
This course covers the following table of contents:
1 Basic Notions
1.1 Probability Space
1.1.1 Sample Space
1.1.2 Events and Probability
1.1.3 Random Variables
1.1.4 Distribution Functions
1.1.5 Basic Distributions
1.1.6 Independent Random Variables
1.1.7 Expectation
1.1.8 Radon-Nikodym's Theorem
1.1.9 Conditional Expectation
1.1.10 Inequalities of Random Variables
1.1.11 Limits of Sequences of Random Variables
1.2 Properties of Limits
1.3 Stochastic Processes
2 Useful Stochastic Processes
2.1 The Brownian Motion
2.2 Geometric Brownian Motion
2.3 Integrated Brownian Motion
2.4 Exponential Integrated Brownian Motion
2.5 Brownian Bridge
2.6 Brownian Motion with Drift
2.7 Bessel Process
2.8 The Poisson Process
2.8.1 Definition and Properties
2.8.2 Interarrival times
2.8.3 Waiting times
3 Properties of Stochastic Processes
3.1 Hitting Times
3.2 Limits of Stochastic Processes
3.3 Convergence Theorems
3.3.1 The Martingale Convergence Theorem
3.3.2 The Squeeze Theorem
4 Stochastic Integration
4.0.3 Non-anticipating Processes
4.0.4 Increments of Brownian Motions
4.1 The Ito Integral
4.2 Examples of Ito integrals
4.2.1 The case Ft = c, constant
4.2.2 The case Ft = Wt
4.3 The Fundamental Relation
4.4 Properties of the Ito Integral
4.5 The Wiener Integral
4.6 Poisson Integration
4.6.1 An Workout Example: the case Ft = Mt
5 Stochastic Differentiation
5.1 Differentiation Rules
5.2 Basic Rules
5.3 Ito's Formula
5.3.1 Ito's formula for diffusions
5.3.2 Ito's formula for Poisson processes
6 Stochastic Integration Techniques
6.0.4 Fundamental Theorem of Stochastic Calculus
6.0.5 Stochastic Integration by Parts