Request for New Course
EASTERN MICHIGAN UNIVERSITY
DIVISION OF ACADEMIC AFFAIRS
REQUEST FOR NEW COURSE
DEPARTMENT/SCHOOL:
CONTACT PERSON:
CONTACT PHONE:
MATHEMATICS________________ COLLEGE:
ARTS AND SCIENCE
DR. OVIDIU CALIN________________________
487-1444
REQUESTED START DATE:
CONTACT EMAIL:
OCALIN@EMICH.EDU
TERM__FALL_______YEAR__2012 _
A. Rationale/Justification for the Course
We first offered this course as a special topics course on stochastic differential and integral calculus in the spring 2011.
It attracted a good number of students with majors in both mathematics and economics and was very positively
received.
The course contains basic notions of probability spaces and random variables, their convergences, and conditional
expectations. It also deals with a presentation of basic stochastic processes, such as Brownian motion, Bessel process,
Poisson process and their main properties. A short chapter deals with stopping and hitting times of a stochastic process.
A good amount of time is allocated to stochastic integration and integration techniques in the sense of Ito, Wiener and
Poisson. Another central topic is stochastic differentiation, differential stochastic equations and solving techniques. We
also cover central theorems of stochastic calculus such as: The Martingale Convergence Theorem, Levy’s
characterization Theorem, Radon-Nikodym's Theorem, and Girsanov's Theorem.
B. Course Information
Math 530
1. Subject Code and Course Number:
2. Course Title:
STOCHASTIC CALCULUS
3. Credit Hours:
3
4. Repeatable for Credit? Yes_______
No__X____
If “Yes”, how many total credits may be earned?_____3__
5. Catalog Description (Limit to approximately 50 words.):
An introduction to stochastic processes, stochastic integration and differentiation, solving stochastic differential
equations, martingale calculus, and martingale measures. A central role is played by the Brownian process and
Poisson processes. A background in differential equations and theoretical probability is assumed.
6. Method of Delivery (Check all that apply.)
a. Standard (lecture/lab)
On Campus
X
X
Off Campus
b. Fully Online
c. Hybrid/ Web Enhanced
7. Grading Mode:
Normal (A-E)
X
Credit/No Credit
8. Prerequisites: Courses that MUST be completed before a student can take this course. (List by Subject Code, Number and Title.)
None
Miller, New Course
Sept. 09
New Course Form
9. Concurrent Prerequisites: Courses listed in #5 that MAY also be taken at the same time as a student is taking this course. (List by Subject
Code, Number and Title.)
None
10. Corequisites: Courses that MUST be taken at the same time as a student in taking this course. (List by Subject Code, Number and Title.)
None
11. Equivalent Courses. A student may not earn credit for both a course and its equivalent. A course will count as a repeat if an equivalent
course has already been taken. (List by Subject Code, Number and Title)
None
12. Course Restrictions:
a. Restriction by College. Is admission to a specific College Required?
College of Business
Yes
No
X
College of Education
Yes
No
X
b. Restriction by Major/Program. Will only students in certain majors/programs be allowed to take this course?
Yes
No
X
If “Yes”, list the majors/programs
c. Restriction by Class Level Check all those who will be allowed to take the course:
Undergraduate
Graduate
All undergraduates_______
All graduate students
X
Freshperson
Certificate
X
Sophomore
Masters
X
Junior
Specialist
Senior
Doctoral
Second Bachelor__ ______
UG Degree Pending__X___
Post-Bac. Tchr. Cert._____
Low GPA Admit_______
Note: If this is a 400-level course to be offered for graduate credit, attach Approval Form for 400-level Course for Graduate
Credit. Only “Approved for Graduate Credit” undergraduate courses may be included on graduate programs of study.
Note: Only 500-level graduate courses can be taken by undergraduate students. Undergraduate students may not register for
600-level courses
d. Restriction by Permission. Will Departmental Permission be required?
Yes
No
(Note: Department permission requires the department to enter authorization for every student registering.)
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Sept. ‘09
X
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New Course Form
13. Will the course be offered as part of the General Education Program?
Yes
No
X
If “Yes”, attach Request for Inclusion of a Course in the General Education Program: Education for Participation in the Global
Community form. Note: All new courses proposed for inclusion in this program will be reviewed by the General Education Advisory
Committee. If this course is NOT approved for inclusion in the General Education program, will it still be offered?
Yes
No
C. Relationship to Existing Courses
Within the Department:
14. Will this course will be a requirement or restricted elective in any existing program(s)? Yes
No
X
If “Yes”, list the programs and attach a copy of the programs that clearly shows the place the new course will have in the curriculum.
Program
Required
Program
15. Will this course replace an existing course? Yes
No
Restricted Elective
Required
Restricted Elective
Yes
No
X
16. (Complete only if the answer to #15 is “Yes.”)
a. Subject Code, Number and Title of course to be replaced:
b. Will the course to be replaced be deleted?
17. (Complete only if the answer #16b is “Yes.”) If the replaced course is to be deleted, it is not necessary to submit a Request for
Graduate and Undergraduate Course Deletion.
a. When is the last time it will be offered?
Term
Year
b. Is the course to be deleted required by programs in other departments?
Contact the Course and Program Development Office if necessary.
Yes
No
c. If “Yes”, do the affected departments support this change?
Yes
No
If “Yes”, attach letters of support. If “No”, attach letters from the affected department explaining the lack of support, if available.
Outside the Department: The following information must be provided. Contact the Course and Program Development office for
assistance if necessary.
18. Are there similar courses offered in other University Departments?
If “Yes”, list courses by Subject Code, Number and Title
Yes
No
X
19. If similar courses exist, do the departments in which they are offered support the proposed course?
Yes
No
If “Yes”, attach letters of support from the affected departments. If “No”, attach letters from the affected department explaining the lack of
support, if available.
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Sept. ‘09
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New Course Form
D. Course Requirements
20. Attach a detailed Sample Course Syllabus including:
a.
b.
c.
d.
e.
f.
g.
h.
Course goals, objectives and/or student learning outcomes
Outline of the content to be covered
Student assignments including presentations, research papers, exams, etc.
Method of evaluation
Grading scale (if a graduate course, include graduate grading scale)
Special requirements
Bibliography, supplemental reading list
Other pertinent information.
NOTE: COURSES BEING PROPOSED FOR INCLUSION IN THE EDUCATION FOR PARTICIPATION IN THE GLOBAL
COMMUNITY PROGRAM MUST USE THE SYLLABUS TEMPLATE PROVIDED BY THE GENERAL EDUCATION
ADVISORY COMMITTEE. THE TEMPLATE IS ATTACHED TO THE REQUEST FOR INCLUSION OF A COURSE IN THE
GENERAL EDUCATION PROGRAM: EDUCATION FOR PARTICIPATION IN THE GLOBAL COMMUNITY FORM.
E. Cost Analysis (Complete only if the course will require additional University resources.
Fill in Estimated Resources for the
sponsoring department(s). Attach separate estimates for other affected departments.)
Estimated Resources:
Year One
Year Two
Year Three
Faculty / Staff
$_________
$_________
$_________
SS&M
$_________
$_________
$_________
Equipment
$_________
$_________
$_________
Total
$_________
$_________
$_________
F. Action of the Department/School and College
1. Department/School
Vote of faculty: For ____16______
Against ____0______
Abstentions ____0______
(Enter the number of votes cast in each category.)
Department Head/School Director Signature
Date
2. College/Graduate School
A. College
College Dean Signature
Date
B. Graduate School (if Graduate Course)
Graduate Dean Signature
Date
G. Approval
Associate Vice-President for Academic Programming Signature
Miller, New Course
Sept. ‘09
Date
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New Course Form
Math 530: Stochastic Calculus
Syllabus
Catalog Description
An introduction to stochastic processes, stochastic integration and differentiation, solving stochastic differential
equations, martingale calculus, and martingale measures. A central role is played by the Brownian process and
Poisson processes. A background in differential equations and theoretical probability is assumed.
Course Goals
The first part of the course contains basic notions of probability spaces and random variables, their convergences,
and 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. Another chapter deals with hitting times, which is
a particular case of stopping times.
A good amount of time is allocated to stochastic integration and integration techniques in the sense of Ito, Wiener
and Poisson. Another central topic is stochastic differentiation, which is covered in detail on two chapters. As an
application, we continue with stochastic differential equations and solving techniques. The first part ends with a
presentation of martingales and martingale measures. In this part, besides important material, we also cover central
theorems of stochastic calculus such as: The Martingale Convergence Theorem, Levy’s characterization Theorem,
Radon-Nikodym's Theorem, and Girsanov's Theorem.
Topics Covered
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
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2.8.3 Waiting times
2.8.4 The Integrated Poisson Process
2.8.5 The Fundamental Relation
2.8.6 The Relations dt dMt = 0, dWt dMt = 0
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
5.3.3 Ito's multidimensional formula
6 Stochastic Integration Techniques
6.0.4 Fundamental Theorem of Stochastic Calculus
6.0.5 Stochastic Integration by Parts
6.0.6 The Heat Equation Method
7 Stochastic Differential Equations
7.1 Definitions and Examples
7.2 Finding Mean and Variance
7.3 The Integration Technique
7.4 Exact Stochastic Equations
7.5 Integration by Inspection
7.6 Linear Stochastic Equations
7.7 The Method of Variation of Parameters
7.8 Integrating Factors
7.9 Existence and Uniqueness
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New Course Form
8 Martingales
8.1 Examples of Martingales
8.2 Girsanov's Theorem
Evaluation
Evaluation will be based upon weekly problem sets, projects, and exams.
A 95-100 %
A- 90 - 94 %
B+ 85 - 89 %
B 80-84 %
B- 75 - 79 %
C+ 70 -74%
C 65 - 70 %
C- 60 – 64%
E below 60%
Textbook
The material for this course will be posted in pdf format on the course webpage.
http://people.emich.edu/ocalin/
Bibliography
1.
2.
3.
4.
Basic Stochastic Processes by Z. Brzezniak and T. Zastawniak, Springer Verlag 2000.
Stochastic Differential Equations by B. Oksendal, Universitext 1979.
Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve, Springer Verlag 1991.
Stochastic Processes by S. Ross, Wiley 1995.
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