SDEs: an introduction and applications

Instructor:     Fei Lu
Class meets:    TTh, 10:30-11:45, Hodson Hall 216
Office Hours: TTh 9:30--10:30,    Krieger 218 
Webpage:          http://www.math.jhu.edu/~feilu/21Fall/SDE21Fall.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

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.

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 three stages: a preliminary plan (due by week 5, on Sep.30, 5%), an oral presentation (in week12--14, 7%) and a final report (due on Nov.30, 8%).

Announcements:

•   COVID and remote learning (tentative): You are encouraged to come for in-person classes following the university's guidance. For those who cannot attend class due to COVID, recording of lectures will be provided.
•   9/3: 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.
•   9/9: Optional coding homework will be added starting from Week 3.
•   10/19: Please pick a date for your project presentation at schedule
  
  Tentative schedule (will be updated weekly):
  

  week	Topics	Homeowork	Due date
  1: 8/31,9/2	Chp2: Math preliminary     LectureNote	2.8, 2.16, 2.17	9/7 solution solution by Ao Sun
  2: 9/7, 9	chp3 Ito and Stratonovich integrals     LectureNote	3.1; 3.4(iii); 3.7ab;	9/14 solution
  3: 9/14, 16	chp4: Ito formula martingale representation     LectureNote	4.4;4.6;4.15 project     .tex	9/21 solution by Joshua Agterberg
  4: 9/21, 23	chp5: SDE     LectureNote	5.1(ii) (iii); 5.7; 5.16(c) Coding: read [Higham01]	9/28
  5: 9/28, 30	SDE and numerical integration Further reading: [KP92] [Higham01] SDE code@FSU     LectureNote	hw5.pdf hw5.tex	10/5 solution by Joshua Agterberg by George Kevrekidis
  6: 10/5, 7	chp6: filtering Further reading: [LSZ15] LectureNote	6.1; 6.2	10/12 solution by Ao Sun by Joshua Agterberg
  7: 10/12, 14	7.1-5 diffusion process LectureNote	7.2(c); 7.4; 7.18	10/19 solution by Joshua Agterberg by Ao Sun
  8: 10/19, 21	diffusion theory: Fokker-Planck equations, Feynman-Kac martingale problem LectureNote Please pick a date for your project presentation at schedule	8.2; 8.6;	10/26 solution by George Kevrekidis by Ao Sun
  9: 10/26, 28	Boundary value problems; LectureNote
  10: 11/2, 4	Girsanov theorem and inference; LectureNote		Ao Sun's slides
  11: 11/9, 11	Langevin Equation (Pav14, Chapter 6) / presentation LectureNote		Qingci An's slides Hao Quan's slides
  12: 11/16, 18	Stochastic Control/ Presentation LectureNote
  13: 11/23, 25	Thanksgiving holiday
  14: 11/30, 12/2	Linear response theory (Pav14, Chapter 9) / Presentation