MA451-F2020 outline - fdsaa PDF

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Fall Term, 2020

MATHEMATICS 451 – Introduction to Stochastic Calculus Department of Mathematics, Faculty of Science, Wilfrid Laurier University

Course instructor: Dr. G. (Joe) Campolieti e-mail: gcamp oli @w lu. ca

Office: LH3073 (Lazaridis Hall)

Office Hours: Tuesdays 1:00 – 2:00 pm. The instructor will schedule these office hours as weekly open live Zoom invitations. Any student requiring help with the course topics can attend these Zoom meetings. Course TA: Mr. Hiromichi Kato (e-mail: [email protected]) The TA will also schedule one-hour open live weekly Zoom meetings for practice problem sessions. Such sessions will allow the TA to further explain solutions to assignments, quizzes and the midterm exam. Lectures: All lectures will be virtual and asynchronously pre-recorded using Virtual Classroom and/or Zoom within My Learning Space. All recorded lectures will be viewable in My Learning Space for every Tuesday and Thursday lecture dates within the semester. Communication: This course uses My Learning Space, https://mylearningspace.wlu.ca/. Material related to the course, including lectures, homework, handouts, assignments, test information, etc., will be posted there. Students are responsible for checking My Learning Space on a regular basis for important announcements. Required Textbook: Financial Mathematics: A Comprehensive Treatment, by G. Campolieti and R.N. Makarov, Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Taylor & Francis Group, 2014. Some Reference Texts: 1. Elementary Stochastic Calculus with Finance in View (Advanced Series on Statistical Science & Applied Probability, Vol. 6), by Thomas Mikosch, World Scientific, Reprinted in 2004. 2. Introduction to Stochastic Calculus with Applications, by Fima C. Klebaner, Imperial College Press, Second, 2005. The third edition is also available. 3. Basic Stochastic Processes: Course through Exercises, by Zdzislaw Brzezniak, Tomasz Zastawniak, Springer, Reprinted in 2007. 4. A First Course in Stochastic Processes, by S. Karlin and H. M. Taylor, 2nd Edition, Academic Press, 1975. 5. Introduction to Stochastic Integration, by Hui-Hsiung Kuo, Springer, 2006. 6. Introduction to Stochastic Integration, Chung, K. L, Williams, R. J, Birkhauser, Second edition, 2013. Calendar Description: Conditional expectations, sigma-algebras, and filtrations; martingales and stopping times; Gaussian processes and Brownian motion; stochastic integration and Ito’s formula; diffusion processes and stochastic differential equations; the Feynman-Kac theorem. Prerequisites: MA250 and ST359 (or MA340)

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Course Learning Outcomes: The student will solidify their formal knowledge on the connection between random variables, sigma-algebras, filtrations and conditional expectations. In particular, the student will learn how to compute conditional expectations by exploiting some useful mathematical identities such as the Independence Proposition. The student will also build a working knowledge of a martingale process, stopping times and a stopped process. The student will develop a basic conceptual and mathematical foundation for the theory and application of Brownian motion and other related diffusion processes. The student will learn how to apply known analytical and probabilistic tools for computing specific conditional expectations that are of importance in continuous-time stochastic modelling. In particular, the student will learn how to apply a main result of the Feynman-Kac formulation for computing several conditional expectations associated to diffusions. The student will also develop the ability to evaluate a certain class of stochastic (Ito) integrals and to solve certain types of stochastic differential equations. The student will also learn to apply various versions of the Ito formula which is a core topic of Ito calculus. The course will also give the student the opportunity to learn a portion of some more advanced topics such as probability measure changes and how certain basic tools of stochastic calculus (e.g., Girsanov’s Theorem) can be used constructively to compute conditional expectations. Multidimensional Brownian motion will also be introduced. As a whole, the theory and specific mathematical tools learned in this course will serve as a very important foundation for latter courses in financial modelling and derivative pricing. Specific Course Topics Covered: 1. Review of sigma algebras of sets; Filtrations; Conditioning on sigma algebra, conditional probabilities and expectations; Martingales; Stopping times. The binomial pricing model and random walks will be used to introduce these topics for discrete-time stochastic processes. (approximate no. of weeks: 1.5) 2. Continuous-time stochastic processes: Markov processes and the transition CDF and PDF; Gaussian processes; Brownian motion and its main properties; Brownian bridge process; geometric Brownian motion; Ornstein-Uhlenbeck process and other simple diffusion processes. Joint distributions of Brownian motion and its maximum or minimum, first-hitting times for Brownian motion and geometric Brownian motion, arc-sine law for Brownian motion. (approximate no. of weeks: 3) 3. Riemann-Stieltjes integral of Brownian motion; Stochastic Itô integration; Itô formulae for functions of Brownian motion and for functions of Itô processes. (approximate no. of weeks: 3) 4. Solutions to linear SDEs; Strong and Weak solutions to Stochastic Differential Equations (SDEs); Feynman-Kac theorems and their applications. (approximate no. of weeks: 2.5) 5. Change of probability measure: Girsanov’s theorem. (approximate no. of weeks: 1) 6. Some topics in Stochastic Calculus for multidimensional Brownian motion. (approximate no. of weeks: 1) Recommended Problems: You should attempt all recommended problems as soon as possible. Practice at problem solving is essential for success in the course! Assignments: All assignments and solutions to assignments will be posted on MyLearningSpace.

Assignments submitted after the due date will be subject to a 20-percentage point penalty per day. Some assignments may carry a stricter late-penalty rule. In such exceptional cases, all students will be made aware of this as part of the assignment. Students will be required to hand in their group assignments as scanned pdf files and uploaded to the appropriate dropbox folders set up on My Learning Space. Students will be notified of the specific folders set up for each assignment. Students shall work in groups of 3-4 with at most four, as specified below. Quizzes: All quizzes will be written remotely during scheduled class times. The TA for the course will notify the students of all scheduled bi-weekly 15-minute (timed) quizzes: 10 minutes for answering plus 5 minutes for scanning and uploading the scanned PDF file. These will be made accessible on My Learning Space. Students will be required to hand in their individual hand-written solutions as a scanned PDF file uploaded to the specific quiz dropbox folder on My Learning Space. Exam Format: All exams shall be written remotely. There will be one midterm test and one final exam on the dates specified below. These will be proctored with the use of Respondus Lockdown Browser (RLDB) + Monitor. More specific details about the exams shall be communicated to the student well in advance of the exam dates. The student is expected to solve exam questions on paper which is then scanned as a PDF file and submitted to the specific My Learning Space folder at the end of the exam. The student can use any scanning device such as a scanner or a phone equipped with scanning software. The student will also require one laptop for both displaying the exam questions and video recording the student during the exam. The camera may be integrated or external. The setup of your webcam (built-in or external) must keep enough of your upper body in view so that your head, hands and your working surface are always visible, and such that you cannot accidentally disconnect it during the exam sessions. The view must be clear enough to see that only the allowed items are on the desk. More detailed instruction will be given prior to the exams. If a technical issue arises that prevents you from writing the exam, it is important to immediately email [email protected]. They will be able to provide technical support, and, if your issue can't be resolved, they will have a record of the incident. If your issue can't be resolved and you are unable even to begin the exam, then you will need to apply for a deferred exam. In the rare event of a catastrophic technical failure during the exam, so that the exam is no longer visible, you should immediately scan and upload your work, and contact your instructor as soon as possible. If agreed to by both you and your instructor, your exam grade will be calculated based on your actual submission. Otherwise, you can petition for a second attempt. In some cases a student might submit a completed exam, but a network issue caused the audio/video monitoring data to be incomplete. If this occurs, the student will be notified by email by the instructor, and the exam will be invalidated. Student Evaluation There will be bi-weekly quizzes, four assignments, one midterm test, and a final exam during the final exam period. The allocation of grades is as follows: Evaluation

Allocation

Quizzes

10%

Assignments

20%

Midterm Test Final Examination (comprehensive)

30% 40%

Schedule

Bi-weekly (held during scheduled calendar class times) 4 in total (students will work in groups of 3-4, with a maximum of 4) October 29, Thursday, 1:00-2:30 pm (2.5 hours, date TBA)

Total

100%

The final mark will be converted to a letter grade in accordance with the conversion table in the undergraduate calendar. NOTE: Students must achieve a score of at least 40% of the marks available on the final examination to be eligible to pass the course.

Course Regulations and Policies

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Specific topics covered on the Tests and the Final Examination will be announced in class and on My Learning Space. Students who have an academic conflict with a test should contact the instructor at least a week in advance of the test. Special arrangements may be possible. A midterm test missed without a documented legitimate (e.g. medical) excuse will be assigned a grade of zero. Any missed quiz will be assigned a grade of zero. The weight of the midterm test missed for legitimate (documented) reasons will be transferred to the Final Examination. The student must submit documentation as soon as possible. Any apparent error in the marking of a test must be reported to the instructor within one week of the date the marked test was returned in class. Marks cannot be adjusted unless this condition is met. If an error in marking is suspected, the instructor may remark the entire test. The graded assignments, quizzes and midterm exam will be made available on My Learning Space. Grades will not be discussed by email or telephone.

Notes Students are expected to be aware of and abide by all University regulations and policies, as outlined in the current Academic Calendar. In particular, 1. Final Examinations Students must reserve the examination period of December 12-23, 2020. If you are considering registering for a special examination or event, you should select a time outside the examination period. Consult Academic Regulations in the Academic Calendar for special circumstances for examination deferment. 2. Special Needs Students with disabilities or special needs are advised to contact Laurier’s Accessible Learning Centre for information regarding its services and resources. Students are encouraged to review the Academic Calendar for information regarding all services available on campus. 3. Academic Integrity/Misconduct (cheating, plagiarism) The University has a defined policy with respect to academic misconduct; penalties are severe and enforced at all times. You are responsible for familiarizing yourself with this policy and accompanying penalty guidelines, and are cautioned that, in addition to failure in a course, students may be suspended or expelled from the University for academic misconduct, and the offence may appear on their transcripts. The relevant policy can be found at Laurier's academic integrity website along with resources to educate and support you in upholding a culture of integrity. Ignorance of Laurier’s academic misconduct policy is not a defence.

Academic misconduct includes, but is not limited to, transmission or reception of information, or possession of unauthorized information, during laboratories, quizzes, tests, or examinations. Academic misconduct also includes plagiarism. Wilfrid Laurier University uses software that can check for plagiarism. If requested to do so by the instructor, students are required to submit their written work in electronic form and have it checked for plagiarism. WLU Policy 12.2 Student Code of Conduct and Discipline provides information on academic and research misconduct code, and the procedures for investigating and determining appropriate disciplinary measures for breaches of this Code. 4. Classroom Use of Electronic Devices The use of electronic devices in the classroom is governed by WLU Policy 9.3: Classroom Use of Electronic Devices. Details of this Policy and the consequences of breaches are stated in the Academic Calendar. Mobile devices such as laptops and tablets may be used in class only for educational (learning) purposes directly related to the course. At times, the instructor may explicitly permit students to use a mobile device to complete an activity or task, at other times, the instructor may ask students to close laptops and turn off tablets in order to focus attention on other course-related tasks. Students who fail to comply may be asked to stow their devices at the front of the classroom, or to leave the classroom.

5. Multi-campus Resource • Good2Talk is a postsecondary school helpline that provides free, professional and confidential counselling support for students in Ontario. Call 1-866-925-5454 or through 2-1-1. Available 24-7. Kitchener/Waterloo Resources • Waterloo Student Food Bank: All students are eligible to use this service to ensure they’re eating healthy when overwhelmed, stressed or financially strained. Anonymously request a package online 24-7. All dietary restrictions accommodated. •

Waterloo Student Wellness Centre: 519-884-0710 x3146, [email protected] or @LaurierWellness The Centre supports the physical, emotional, and mental health needs of students. Located on the 2nd floor of the Student Services Building, the Centre is operational Monday through Friday, 9 am12 pm and 1-4pm. To book an appointment with a physician, nurse, mental health nurse, or counsellor, call the Centre at 519-884-0710 x3146. Please note, everyone entering the Student Wellness Centre is required to bring their own mask and wear it at all times. After hours crisis support available 24/7. Call 1-844-437-3247 or 519-821-3582 (Here 24/7). Visit Laurier’s Mental Health Resources page under ‘Remote Resources’ for more support during the current disruption to in-person counselling services. If you are experiencing a new or worse cough, shortness of breath, fever, or have had close contact with a confirmed or probably case of COVID-19, go to the Government of Ontario COVID-19 page

Course Drop Dates Fall 2020 September 16: Last day to drop/withdraw from 12-week course(s) at no tuition charge (provided the student remains registered in at least one course).

September 23: Last day to drop/withdraw from 12-week course(s) at 10% tuition charge (assessed at course rate). November 11: Last day to drop 12-week course(s) or withdraw without failure and possible tuition adjustment (tuition charge assessed at 55% of the course rate)....