Math 2604: Advanced Scientific Computing IV
Numerical Solution of Stochastic Differential Equations

Instructor: Catalin Trenchea
Phone: (412) 624-5681
Office: Thackeray 606

Lecture: MWF 1:00-1:50; 213 Cathedral of Learning

Office Hours: Tu 10:00am-noon, Thu 10:00am-noon and by appointment.

Textbook: Peter E. Kloeden, Eckhard Platen Numerical Solution of Stochastic Differential Equations, Springer 1992.

Prerequisites: Good undergraduate background in mathematical methods.
Lectures will adapt to diverse backgrounds. Please contact the instructor if you have questions about your preparation.

Content: The course focuses on the fundamental mathematical aspects of numerical methods for stochastic differential equations, motivated by applications in physics, engineering, biology, economics. It provides a systematic framework for an understanding of the basic concepts and of the basic tools needed for the development and implementation of numerical methods for SDEs, with focus on time discretization methods for initial value problems of SDEs with Ito diffusions as their solutions. The course material is self-contained.

Topics to be covered: Background material on probability, stochastic processes and statistics, introduction to stochastic calculus, stochastic differential equations and stochastic Taylor expansions. The numerical methods for time discretization of ODEs are briefly reviewed, then methods for time discretization for SDEs are introduced and analyzed.

Homework: Written homework will be assigned, and there will be a in class final presentation.

Other references:
Peter Eris Kloeden, Eckhard Platen, Henri Schurz Numerical Solution of SDE Through Computer Experiments, Springer 1994.
Alexandre J. Chorin, Ole H. Hald Stochastic Tools in Mathematics and Science Second Edition, Springer 2009.
Desmond J. Higham An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Rev. 43 (2001), no. 3, 525-546.
Dongbin Xiu Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press 2010.
Max D. Gunzburger Fifteen Ways to Fool the Masses When Presenting your Work in UQ, SIAM News, Volume 45, Number 8, 2012.

The course web page will be updated continuously throughout the semester. The student is responsible for checking this web page for assignments and policies.

Disability Resource Services
If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services, 140 William Pitt Union, 412-648-7890 or 412-383-7355 (TTY) as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.
Academic Integrity
Cheating/plagiarism will not be tolerated. Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction. On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.
Statement on Classroom Recording
To address the issue of students recording a lecture or class session, the University's Senate Educational Policy Committee issued the recommended statement on May 4, 2010. ``To ensure the free and open discussion of ideas, students may not record classroom lectures, discussion and/or activities without the advance written permission of the instructor, and any such recording properly approved in advance can be used solely for the student's own private use."