SDEs: an introduction and applications

Instructor:     Fei Lu
Class meets:    TTh, 10:30-11:45, Hodson Hall 301
Office Hours: TTh 9:30--10:30,    Krieger 218 
Webpage:          http://www.math.jhu.edu/~feilu/22Fall/SDE22Fall.html
Email:             feilu##   ( ## = @math.jhu.edu)

Textbook:  Bernt Oksendal: Stochastic differential equations- an introduction with applications. 6th Edition
Other reference books (some of them have electronic copies available to JHU):

• [Pav14]   Grigorios A. Pavliotis: Stochastic Processes and Applications. Springer 2014. ISBN 978-1-4939-1322-0
• [Kut04]   Kutoyants: Statistical Inference for Ergodic Diffusion Processes, Springer 2004. ISBN 978-1-84996-906-2.
• [KP92]    Peter E. Kloeden and Eckhard Platen: Numerical Solution of Stochastic Differential Equations. Springer, 1992. ISBN 978-3-662-12616-5
• [ELV19]  Weinan E, Tiejun Li, and Eric Vanden-Eijnden. Applied Stochastic Analysis. American Mathematical Society, 2019. ISBN-13: 978-1470449339
• [Kha12]  Rafail Khasminskii: Stochastic stability of differential equations, Springer, 2012. ISBN 978-3-642-23280-0
• [CH13]   Alexandre J. Chorin and Ole H. Hald: stochastic tools in Mathematics and Science, Springer 2013. ISBN 978-1-4419-1002-8
• [LSZ15]  Kody Law, Andrew Stuart and Konstantinos Zygalakis: Data Assimilation: A Mathematical Introduction, Springer 2015. ISBN 978-3-319-20325-6.
• [Higham01] Higham 2001: An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations
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• [KFLOL21]   Patrick Kidger, James Foster, Xuechen Li, Harald Oberhauser, Terry Lyons: Neural SDEs as Infinite-Dimensional GANs

Course Syllabus

Course plan (tentative): This course is an introduction to stochastic differential equations and applications. Basic topics to be reviewed include Ito and Stratonovich integrals, Ito formula, SDEs and their integration. The course will focus on diffusion processes and diffusion theory, with topics include Markov properties, generator, Kolmogrov's equations (Fokker-Planck equation), Feynman-Kac formula, the martingale problem, Girsanov theorem, stability and ergodicity. The course will briefly introduce applications, with topics include statistical inference of SDEs, filtering and control, signature method (if time permits).

Prerequisite: familiar with graduate level probability, real analysis and PDE. Exposure to measure theory and functional analysis will be a plus.

Grading: Grade will be based on homework assignments and a project. Homework  (80%); Presentation (20%)   The purpose of homework/project is to help learning. Please let me know if the workload is unmanageable, and I will make adjustments so that the workload is manageable. Cheating (both parties) will be penalized.

• There will be 8 homework assignments in the first 8 weeks. They will be assigned on each Thursday and due on the next Tuesday.    
• Project: the project can be any topic closely related to the course, e.g., parts of your research topic, review of a topic, or numerical simulations. The project consists of two stages: a preliminary plan (due by week 10, on Nov.3), an oral presentation or report (in week12--14, due Dec.1) .

Announcements:

•   This course is a graduate course, so it has "0" credits for graduate students. Please contact your department if you would like to have credits.

Tentative schedule (will be updated weekly):


week	Topics	Homeowork	Due date
1: 8/30,9/1	Chp2: Math preliminary     LectureNote	2.8, 2.16, 2.17	9/6
2: 9/6, 8	chp3 Ito and Stratonovich integrals     LectureNote	3.1; 3.4(iii); 3.7ab;	9/13
3: 9/13, 15	chp4: Ito formula martingale representation     LectureNote	4.4;4.6;4.15	9/20
4: 9/20, 22	chp5: SDE     LectureNote	5.1(ii) (iii); 5.7; 5.16(c)	9/27
5: 9/27, 29	SDE and numerical integration Further reading: [KP92] [Higham01] SDE code@FSU     LectureNote	hw5.pdf hw5.tex	10/4
6: 10/4, 6	chp6: filtering Further reading: [LSZ15] LectureNote	6.1; 6.2	10/11
7: 10/11, 13	7.1-5 diffusion process LectureNote	7.2(c); 7.4; 7.18	10/18
8: 10/18, 20	diffusion theory: Fokker-Planck equations, Feynman-Kac LectureNote	8.2; 8.6;	10/25
9: 10/25, 27	the martingale problem Girsanov theorem and inference;
10: 11/1, 3	Girsanov theorem and inference; LectureNote
11: 11/8, 10	Gradient systems (Pav14, Chapter 4.5); Langevin Equation (Pav14, Chapter 6) LectureNote
12: 11/15, 17	Boundary value problems; Stochastic Control/ BSDE LectureNote
13: 11/22, 24	Thanksgiving holiday
14: 11/29, 12/1	Stochastic Control / Linear response theory(Pav14, Chapter 9) LectureNote
14: 12/6, 12/8	Linear response theory / Neural SDE/ Presentations