Stochastic Processes

Introduction to Stochastic Calculus (MATH 545, Spring 2020)

Meeting: Physics 119, Mon-Wed 4:40-5:55 p.m., replacement lectures: TBA

Office hours: 12:00-2:00pm on TUE in Gross Hall 359 (subject to change) and by appointment,

E-mail: agazzi at math.duke.edu (please include MATH 545 in your email title).

Couse Description: This is an introductory, graduate-level course in stochastic calculus and stochastic differential equations, oriented towards topics that have applications in the natural sciences, engineering, economics and finance. A tentative schedule of topics is given below.

Prerequisites: Real analysis (MATH 431) and Probablity (MATH 230 or MATH 340). If you have not done well in these courses, you should consult the instructor before enrolling in this class. Some exposure to measure theory is a plus (via MATH 641 or equivalent), but it is not strictly required.

Sakai Website: All information on this course, syllabus, lecture notes, and homework assignments will also be progressively posted on the Sakai website. Log in with your NetID, and look for files under the "Resources" tab for our class.
You are encouraged to use the piazza website to ask questions about homework and class material; that way, the entire class can benefit from each question or comment.

Textbooks and references: The Principal Reference (required) of this course is

Introduction to Stochastic Calculus with Applications. Fima C. Klebaner, Imperial College Press, 2012. 3rd edition.

Here are some other useful texts, some of which are available in the library:

Stochastic Differential Equations. B. Oksendal, 6th edition, Springer 2013.
An Introduction to Stochastic Differential Equations. L. Evans, AMS, 2014.
Introduction to the Theory of Random Processes. I. Gikhman, A. Skorokhod, W. B. Saunders Company 1963.
Continuous Martingales and Brownian Motion. D. Revusz, M. Yor, Springer 1990.
Stochastic calculus: A practical introduction. R. Durrett, CRC Press, 1996.

Some of these books are available at the library. Some unofficial lecture notes are available for download here. These noes will be periodically updated during the course and are not its main reference.

Grading: Judgement based on Final (30%) and Midterm (20%) exam marks and on consistent Homework efforts (50%).

Homework: HW0 (due JAN 8th), HW1 (due JAN 15th), HW2 (due JAN 27th) HW3 (due FEB 17th).
Solutions: HW1_sol, HW2_sol.

Please deliver your assignment during the class on the due date (late homework solutions will not be graded). Homework should be stapled, coherent and legible. Collaboration is allowed in solving the problems, but you are to provide your own independently written solution; copying answers is not allowed.

Reading: Reading assignments will be given with problem sets. You are expected to complete reading assignments before the corresponding class.

Tests: We will have one in-class midterm and one final exam. These tests will cover all class material, including ideas introduced in the homework problems, in the readings, and in class meetings. Taking the final is a necessary requirement for passing the class. No collaboration is allowed on exams. If you have to miss the in-class midterm, please arrange with the instructor at least a week ahead. In this case, you can only choose to take the exam earlier than the scheduled time.

Midterm: TBA, Room: TBA.

Final: TBA

Participation: Attendance to all class meetings is required.

Supplementary material: There is a good collection of problems posted on this website

Tentative Schedule of Topics (to be updated):

Introduction and probabilistic background, conditional expectations
Brownian motion and its properties
Stochastic integration and Itô processes
Stochastic differential equations (SDEs)
Stopping times and Martingales
Change of measure, Girsanov's theorem
Diffusion Processes
Applications in Finance: Black-Scholes equation
Pure Jump Processes (if time permits)
Applications in Biology (if time permits)
Stochastic Control (if time permits)