SAMPLE COURSE OUTLINE CMTH 500 INTRODUCTION TO STOCHASTIC

SAMPLE COURSE OUTLINE
CMTH 500
INTRODUCTION TO STOCHASTIC
PROCESSES
This is a sample course outline only. It should not be used to plan assignments or purchase textbooks.
A current version of the course outline will be provided by the instructor once the course begins.
Every effort will be made to manage the course as stated. However, adjustments may be necessary at the
discretion of the instructor. If so, students will be advised and alterations discussed in the class prior to
implementation.
It is the responsibility of students to ensure that they understand the University’s policies and procedures,
in particular those relating to course management and academic integrity
COURSE DESCRIPTION
The following topics within Stochastic Calculus will be mastered: Introduction to probability spaces.
Conditional expectation; Discrete time martingales, Poisson Process; Martingales in continuous time and
Brownian motion; Stochastic integration. Ito’s lemma and introduction to stochastic differential
equations.
This course is designed to provide students with an understanding of Stochastic Calculus with
applications in Finance.
COURSE OBJECTIVE/LEARNING OUTCOMES
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
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To provide a sound foundation in probability.
To provide examples of basic discrete and continuous-time stochastic processes.
To provide students with a workable knowledge of Ito’s lemma, Feynman-Kac theorem and its
applications in finance.
CORE TOPICS:
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Sigma–algebras of sets. Filtrations. Conditional probabilities and expectations.
Discrete time processes. Random walks. Markov Process.
Martingales. Stopping times. Poisson process. Brownian motion.
Stochastic integrals. Itô’s Lemma. Geometric Brownian motion. Diffusion processes.
The Feynman-Kac theorem and its applications.
Sample Course Outline
Introduction to Stochastic Processes
Fall 2012
Page 1 of 5
CMTH 500
TEXTBOOK AND READING LISTS
This is a sample course outline only. It should not be used to purchase textbooks. A current version
of the course outline will be provided by the instructor once the course begins.
Required Text:
[1] - Introductory Stochastic Analysis for Finance and Insurance, X. Sheldon Lin, Wiley
Series in Probability and Statistics.
Readings and Related Material:
Elementary Stochastic Calculus With Finance in View (Advanced Series on Statistical science & Applied
Probability, Vol. 6), by Thomas Mikosch, World Scientific, Reprinted 2004.
Introduction to Stochastic Calculus with Applications, by Fima C. Klebaner, Imperial College Press,
Second edition 2005.
Basic Stochastic Processes: Course through Exercises, by Zdzislaw Brzezniak, Tomasz Zastawniak,
Springer, Reprinted 2007.
A First Course in Stochastic Processes, by S. Karlin and H. M. Taylor, 2nd Edition, Academic Press.
Stochastic Processes with Applications to Finance, Masaaki Kijima
Introduction to Stochastic Calculus Applied to Finance. Damien Lamberton and Bernard Lapeyre
Steven E. Shreve " Stochastic Calculus for Finance" parts I and II.
COURSE STRUCTURE AND ORGANIZATION
This is a sample course outline only. A current version of the course outline will be provided by the
instructor once the course begins.
Each class will consist of two components: A lecture that covers theory and an overview of practical
applications of concepts and a demonstration of the concepts learn in financial applications.
Week
Topic
Details
1
Introduction
Introduction to Probability spaces. Probability Review.
Random variables. (1) Ch 2
2
Distributions
Random vectors, distributions and expectations. (1) Ch.2
3
Conditioning
Conditional probability, conditional expectation. (1) ch 2
Sample Course Outline
Introduction to Stochastic Processes
Fall 2012
Page 2 of 5
CMTH 500
4
5
6
Discrete Stochastic
Processes
General introduction discrete-time stochastic processes:
random walk. (1) Ch 3
Assignment 1
Vector value variables and Conditional Expectation.
Discrete Stochastic
Processes
Definition and properties of Martingale. (1) Ch 3
Continuous time
General description of continuous-time stochastic processes.
Assignment 1 Due
Examples of Processes
Definition and properties of Brownian motion and Poisson
Process (1) Ch 4
Stochastic integration
Quadratic variation, covariation, stochastic integration (1) Ch 5
Assignment 2
Martingales, Posson Process
9
Stopping Times
Martinglaes, stopping times, the optional sampling theorem.
(1) Ch 4
10
Ito Integration
Stochastic (Ito) integration. (1) Ch 5
Stochastic Differential
Equations
One dimensional It’s Lemma; the stochastic version of
integration by parts. (1) Ch 5
Assignment 3
Brownian Motion. Ito’s lemma
7
8
11
Assignment 2 Due
12
Feyman – Kac
Exponential martingales, the Feynman-Kac formula (1) Ch. 6
13
Exam
Exam: Multiple Choice/Short Answer
METHOD AND SCHEDULE OF STUDENT EVALUATION
This is a sample course outline only. It should not be used to plan assignments. A current version of the
course outline will be provided by the instructor once the course begins.
3 assignments worth 20% each
Sample Course Outline
Introduction to Stochastic Processes
60%
Fall 2012
Page 3 of 5
CMTH 500
Final examination (non-lab)
40%
Total
100%
Lab Assignments (graded):
Assignment 1: Problems on Vector value variables and Conditional Expectation.
Assignment 2: Problems involving Martingales and Poisson Processes.
Assignment 3: Brownian Motion problems and applications of Ito’s lemma.
Assignments are due at the beginning of the evening set out in the schedule above. The instructor must
approve any extension prior to the due date. No lab assignment submission will be accepted for
grading once graded labs have been returned to students, normally at the next class after being
handed in.
MISSED TERM WORK OR EXAMINATIONS
Students are expected to complete all assignments, tests, and exams within the time frames and by the
dates indicated in this outline. Exemption or deferral of an assignment, term test, or final examination is
only permitted for a medical or personal emergency or due to religious observance. The instructor must
be notified by e-mail prior to the due date or test/exam date, and the appropriate documentation must be
submitted. For absence on medical grounds, an official student medical certificate, downloaded from the
Ryerson website at http://www.ryerson.ca/senate/forms/medical.pdf or picked up from The Chang
School at Heaslip House, 297 Victoria St., Main Floor, must be provided. For absence due to religious
observance, visit http://www.ryerson.ca/senate/forms/relobservforminstr.pdf to obtain and submit the
required form.
PLAGIARISM
The Ryerson Student Code of Academic Conduct defines plagiarism and the sanctions against students
who plagiarize. All Chang School students are strongly encouraged to go to the academic integrity
website at www.ryerson.ca/academicintegrity and complete the tutorial on plagiarism.
ACADEMIC INTEGRITY
Ryerson University and The Chang School are committed to the principles of academic integrity as
outlined in the Student code of Academic conduct. Students are strongly encouraged to review the student
Sample Course Outline
Introduction to Stochastic Processes
Fall 2012
Page 4 of 5
CMTH 500
guide to academic integrity, including penalties for misconduct, on the academic integrity website at
www.ryerson.ca/academic integrity and the Student code of Academic conduct at
www.ryerson.ca/senate/policies.
RYERSON STUDENT EMAIL
All students in full and part-time graduate and undergraduate degree programs and all continuing
education students are required to activate and maintain their Ryerson online identity at
www.ryerson.ca/accounts in order to regularly access Ryerson’s E-mail (Rmail), RAMSS, my.ryerson.ca
portal and learning system, and other systems by which they will receive official University
communications.
COURSE REPEATS:
Senate GPA policy prevents students from taking a course more than three times. For complete GPA
policy see policy no. 46 at www.ryerson.ca/senate/policies.
RYERSON ACADEMIC POLICIES
For more information on Ryerson’s academic policies, visit the Senate website at www.ryerson.ca/senate.
Course Management Policy No. 145
Student Code of Academic conduct No. 60
Student code of non-Academic Conduct No. 61
Examination Policy No. 135
Policy on Grading, Promotion, and Academic Standing Policy No. 46
Undergraduate Academic consideration and Appeals Policy No. 134
Accommodation of Student Religious Observance Obligations Policy no. 150
Sample Course Outline
Introduction to Stochastic Processes
Fall 2012
Page 5 of 5
CMTH 500