Math 110.653 - SDEs: an introduction and applications

Instructor:     Fei Lu
Class meets:    MW, 8:30-9:45,     Bloomberg 276
Office Hours:  MW, 9:45-10:45,    Krieger 218 
Webpage:          http://www.math.jhu.edu/~feilu/26Spring/SDE26Spring.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
• [LSKME2]   Chieh-Hsin Lai, Yang Song, Dongjun Kim, Yuki Mitsufuji, Stefano Ermon: The Principles of Diffusion Models

Course Syllabus

Course plan (tentative): This course is an introduction to stochastic differential equations (SDEs) and applications. Basic elements 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 and generators, Fokker-Planck equation, Feynman-Kac formula, the martingale problem, Girsanov theorem, stability and ergodicity. The course will also briefly introduce applications, with topics include statistical inference of SDEs, filtering and control, neural SDEs and diffusion models (following the interests of the class).

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 too heavy, and I will make adjustments so that the workload is manageable.

• There will be 7 homework assignments in the first 8 weeks. They will be assigned on each Monday 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 12, Wed.), an oral presentation or report (in week14--15) .

Announcements: Any announcements related to the course (e.g., homework assignment, changes of schedule) will be posted on the course webpage. Please check it regularly.

• 20260127: Please turn in homework to Gradescope. You are ecouraged to discuss homework problems with classmates or use AI tools, but the work you turn in must be your own. If you use latex, a template is available here .