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 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.
B. Course Information
Math 430
1. Subject Code and Course Number:
2. Course Title:
INTRODUCTION TO STOCHASTIC CALCULUS
3. Credit Hours:
3
4. Repeatable for Credit? Yes_______
No__X____
If “Yes”, how many total credits may be earned?_____
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.
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.)
MATH 223: Multivariable Calculus
MATH 325: Differential Equations
MATH 370: Probability and Statistics I
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.)
Miller, New Course
Sept. 09
New Course Form
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___X____
All graduate students____
Freshperson
Certificate
Sophomore
Masters
Junior
Specialist
Senior
Doctoral
Second Bachelor__ X______
UG Degree Pending_____
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.)
13. Will the course be offered as part of the General Education Program?
Yes
No
X
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
Miller, New Course
Sept. ‘09
Page 2 of 6
New Course Form
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
X
No
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
mathematics major
Required
Restricted Elective
Program
mathematics minor
Required
Restricted Elective
15. Will this course replace an existing course? Yes
No
X
X
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?
Yes
No
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.
D. Course Requirements
20. Attach a detailed Sample Course Syllabus including:
a.
b.
c.
d.
e.
f.
g.
h.
Miller, New Course
Sept. ‘09
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.
Page 3 of 6
New Course Form
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
Page 4 of 6
New Course Form
Math 430: Introduction to Stochastic Calculus , 3 credit hours
Syllabus
Catalog Description
An introduction to stochastic processes, stochastic integration and differentiation, solving stochastic differential
equations, martingale calculus, and martingale measures.
Pre requisites: MATH 223, MATH 325, and MATH 370
Course Goals
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.
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
2.8.3 Waiting times
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Sept. ‘09
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New Course Form
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
Evaluation
Evaluation will be based upon weekly problem sets and exams.
A 95-100 %
A- 90 - 94 %
B+ 85 - 89 %
B 80-84 %
B- 75 - 79 %
C+ 70 -74%
C 65 - 70 %
D 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.
Miller, New Course
Sept. ‘09
Page 6 of 6
Mathematics Major
Additional Requirements: 3 hours
One course from the following:
COSC 111 - Introduction to Programming 3 hrs
COSC 120 - Computational Principles for Mathematics and the Sciences 3 hrs
Major Requirements: 36 hours
Required Courses: 24 hours
MATH 120 - Calculus I (Gen Ed Area II) 4 hrs
MATH 121 - Calculus II 4 hrs
MATH 122 - Elementary Linear Algebra 3 hrs
MATH 211 - Introduction to Mathematical Proof 3 hrs
MATH 223 - Multivariable Calculus 4 hrs
MATH 370 - Probability and Statistics I 3 hrs
One course from the following:
MATH 411 - Abstract Algebra 3 hrs
MATH 416 - Linear Algebra 3 hrs
MATH 420 - Introduction to Real Analysis 3 hrs
Restricted Elective Courses: 12 hours
Twelve hours from the following:
MATH 311W - Mathematical Problem Solving (Gen Ed Area I, W) 3 hrs
MATH 319 - Mathematical Modeling 3 hrs
MATH 325 - Differential Equations 3 hrs
MATH 341 - College Geometry 3 hrs
MATH 372 - Problems in Actuarial Studies I 2 hrs
MATH 407 - Elementary Number Theory 3 hrs
MATH 409 - Cryptology 3 hrs
MATH 411 - Abstract Algebra 3 hrs
MATH 416 - Linear Algebra 3 hrs
MATH 418 - Modeling with Linear Algebra 3 hrs
MATH 419W - Introduction to Stochastic Mathematical Modeling (Gen Ed Area I, W) 3 hrs
MATH 420 - Introduction to Real Analysis 3 hrs
MATH 424 - Introduction to Complex Variables 3 hrs
MATH 425 - Mathematics for Scientists 3 hrs
MATH 430 – Introduction to Stochastic Calculus 3 hrs
MATH 436 - Introduction to Numerical Analysis 3 hrs
MATH 460 - Applied Survey Sampling 3 hrs
MATH 461 - Linear Regression Analysis 3 hrs
MATH 462 - Design and Analysis of Experiments 3 hrs
MATH 468 - Introduction to Biostatistics 3 hrs
MATH 469 - Introduction to Categorical Data Analysis 3 hrs
MATH 471 - Probability and Statistics II 3 hrs
MATH 472 - Problems in Actuarial Studies II 2 hrs
MATH 474W - Applied Statistics (Gen Ed Area I, W) 3 hrs
Minor Requirements:
This program requires a minor. Please contact your program adviser for a list of possible minors.
Program Total:
Students must earn a minimum total of 124 credits at the 100-level or above.
Notes:
Each student must choose a writing intensive course as part of major completion requirements. Consult
your adviser for course options.
Mathematics Minor
Additional Requirements: 3 hours
One course from the following:
COSC 111 - Introduction to Programming 3 hrs
COSC 120 - Computational Principles for Mathematics and the Sciences 3 hrs
Required Courses: 14 hours
MATH 120 - Calculus I (Gen Ed Area II) 4 hrs
MATH 121 - Calculus II 4 hrs
MATH 122 - Elementary Linear Algebra 3 hrs
One course from the following:
MATH 211 - Introduction to Mathematical Proof 3 hrs
MATH 360 - Statistical Methods 3 hrs
MATH 370 - Probability and Statistics I 3 hrs
Restricted Elective Courses: 6 hours
Six credit hours from the following:
* MATH 205 - Mathematical Structures for Computer Science 4 hrs
MATH 223 - Multivariable Calculus 4 hrs
MATH 311W - Mathematical Problem Solving (Gen Ed Area I, W) 3 hrs
MATH 319 - Mathematical Modeling 3 hrs
MATH 325 - Differential Equations 3 hrs
MATH 341 - College Geometry 3 hrs
** MATH 360 - Statistical Methods 3 hrs
** MATH 370 - Probability and Statistics I 3 hrs
MATH 372 - Problems in Actuarial Studies I 2 hrs
MATH 407 - Elementary Number Theory 3 hrs
MATH 409 - Cryptology 3 hrs
MATH 411 - Abstract Algebra 3 hrs
MATH 416 - Linear Algebra 3 hrs
MATH 419W - Introduction to Stochastic Mathematical Modeling (Gen Ed Area I, W) 3 hrs
MATH 418 - Modeling with Linear Algebra 3 hrs
MATH 420 - Introduction to Real Analysis 3 hrs
MATH 424 - Introduction to Complex Variables 3 hrs
MATH 425 - Mathematics for Scientists 3 hrs
MATH 436 - Introduction to Numerical Analysis 3 hrs
MATH 460 - Applied Survey Sampling 3 hrs
MATH 461 - Linear Regression Analysis 3 hrs
MATH 462 - Design and Analysis of Experiments 3 hrs
MATH 468 - Introduction to Biostatistics 3 hrs
MATH 469 - Introduction to Categorical Data Analysis 3 hrs
MATH 471 - Probability and Statistics II 3 hrs
Minor Total: 23 hours