Stochastic Differential Equations
MATH8008, fall semester, 2023
Instructor: Yizao Wang
Email: yizao.wang@uc.edu
Office: French Hall 4302.
Office Hours: by appointment.
Class meeting in hybrid mode: 2 in-person lectures MW 1:25-2:20pm, 324 Braunstein, and a pre-recorded third lecture each week on Canvas.
Textbook
- Stochastic Differential Equations. An introduction with applications. Øksendal, Bernt. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003.
- (Recommended) Stochastic Calculus and Financial Applications, Michael Steele.
Applications of Mathematics (New York), 45. Springer-Verlag, New York, 2001.
- (Recommended) An Introduction to Stochastic Differential Equations, Lawrence Evans. American Mathematical Society, Providence, RI, 2013.
Course description
This course provides an introduction to stochastic calculus and stochastic differential equations. The lectures will roughly follow Chapter 2, Appendix B, C, Chapters 3-5 and 7-9, of Oksendal's book. If time permits, some research-related topics will be discussed towards the end of the semester.
Measure-theoretic probability (MATH/STAT7032) is required. Markov chains and martingales are strongly (MATH6008) recommended. Related topics/materials will be reviewed in a relatively fast manner at the beginning of the semester.
Grades
- Based on 6-8 homework problem sets. No exams.
Tentative Schedule
- Week 1 (Aug 21, Mon): Chapter 2, Some mathematical preliminaries.
- Week 2 (Aug 28): Appendix B, Conditional expectation.
- Week 3 (Sep 4): No class on Labor Day, Monday Sep 4.
- Week 4 (Sep 11): Appendix C, Uniform integrability and martingale convergence.
- Week 5 (Sep 18):
- Week 6 (Sep 25): Chapter 3, Ito integrals.
- Week 7 (Oct 2): Chapter 4, The Ito formula and the martingale representation theorem.
- Week 8 (Oct 9): Fall Reading Days, no class on Mon Oct 9.
- Week 9 (Oct 16): Chapter 5, Stochastic differential equations.
- Week 10 (Oct 23):
- Week 11 (Oct 30):
- Week 12 (Nov 6): Veterans Day Holiday, no class on Nov 10, Fri.
- Week 13 (Nov 13):
- Week 14 (Nov 20): Thanksgiving Weekend Holiday, no class on Nov 24, Fri.
- Week 15 (Nov 27):
- Week 16 (Dec 4): Exam week.