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    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:      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