Topic outline

  • Course topic, target audience, and prerequisites

    Topic: The course is an introduction to stochastic differential equations (SDEs) from an applied point of view. The contents include the theory, applications, and numerical methods for SDEs. Examples of SDE models are given in mechanics and electrical engineering, physics, target tracking, and machine learning. The course uses flipped classroom distance teaching: before each session, students follow online prerecorded lectures and/or read textbook sections, and work on homework problems. In the virtual contact sessions, the current lecture topic and the homeworks are discussed.

    Target audience: Advanced undergraduate and graduate (PhD) students. Researchers and engineers wishing to get a hands-on introduction to the topic.

    Prerequisites: Multivariate differential and integral calculus, matrix analysis, basic probability, Matlab/Octave/Python.

    Course in Summer 2020

    The course will be arranged by Profs. Simo Särkkä ( and Arno Solin ( in summertime 1.7.-28.8.2020 and the exam is on week 31.8.-4.9. The course assistant is Zheng Zhao (

    The course is fully online although virtual sessions contact sessions are organized via course Slack team weekly. The virtual contact sessions are organized weekly on Wednesdays 14:15-16:00. The first contact session is 1.7. at 14:15-16:00. 

    The course will be arranged as a flipped class type of course:

    • The coursebook is available online here: and the example codes from the book here:

    • Lectures are delivered as YouTube videos (will be linked in the Materials section) which are independently studied by the students before the contact sessions. Some of the lectures are delivered in the form of independent reading instructions instead of videos.

    • Lecture Slides are also available in the Materials section.

    • Quizzes attached to the lectures should be completed as well and completing all of them gives 1 extra homework exercise point. However, the first quiz is mandatory.

    • Contact sessions are organized in Slack such that a lecturer is present at the slack room (Simo Särkkä, Arno Solin, or Zheng Zhao). The contact sessions are discussion sessions about the current lecture/material and the homework. 

    • Homework exercises must be completed preferably before the contact sessions (see Homework exercises and contact sessions section for information) and they must be returned via MyCourses in two batches: rounds 1-4 on 31.7. and rounds 5-8 on 28.8. At least 50% (12/24) of homework exercises must be completed and completing at least 75% (18/24) of exercises gives a +1 grade increase to the exam grade. 

    • The project work topic should be selected by 26.7.2020 (report it here in MyCourses). The deadline of the project work is 28.8.2020 (return it in MyCourses).

    • The course exam will be organized in the week 31.8.-4.9. as an online exam (details communicated directly to attendants, let us know if you didn't get the details). 

    • The grade of the course is the maximum of the exam grade and the project work grade. However, you must get a passing grade from both the exam and project work to pass the course.

  • The contact sessions (Wednesdays 14:15-16:00 in course slack) are discussion sessions of lecture and homework exercises that the students have (preferably) solved beforehand. Recall that the lectures/materials must be studied before the sessions and the quizzes must be completed before them as well! The homework exercises are related to the lectures and hence it is advisable to watch the videos first and then do the exercises. There are two deadlines for returning the exercise answers:

    • Rounds 1-4 on 31.7.

    • Rounds 5-8 on 28.8.

    At least 50% (12/24) of homework exercises must be completed and completing at least 75% (18/24) of exercises gives a +1 grade increase to the exam. Finishing all quizzes gives one extra exercise point.

    The contact session topics are given in the following, the exercises are from the coursebook:

  • The main textbook of the course is

    Simo Särkkä and Arno Solin (2019). Applied Stochastic Differential Equations. Cambridge University Press.

    The book can be found in this link:

    The examples codes of the book can be found here:

    Lecture Slides can also be found down below as well as the YouTube videos and the quizzes.

  • Project work overview

    • Select your favorite topic e.g. from the coursebook among the topics which have not been covered on the course. However, you can also think of an application of SDE methods in your own research or application.
    • Write around 5-10 pages report which contains a brief review of the theory and then a numerical illustration. That is, a short technical report/article on the subject.
    • Deadlines are 26.7.2020 (topic) and 28.8.2020 (final report)

    Project topics

    • Strong and weak convergence of numerical methods
    • Exact simulation of SDEs
    • Lamperti transform
    • Doob’s h-transform
    • Girsanov theorem
    • Variational Bayes approximations of SDEs
    • Series expansions of SDEs
    • Small noise expansion approximations
    • Numerical solution of Fokker-Planck-Kolmogorov equations
    • State-space methods for Gaussian processes
    • Solution of PDEs with Feynman-Kac
    • Wiener/Feynman path integrals and SDEs
    • Existence and uniqueness of SDEs
    • Martingale representation theorem
    • Taylor series expansions of moment equations
    • Levy-process driven SDEs
    • Spatially distributed systems
    • Black-Scholes formulae
    • Physics application
    • Biological application
    • Communications application
    • Navigation application
    • Own topic

    Project work report

    The report should be returned in PDF form below by 28.8.2020 and it should contain at least the following:

    • Introduction. Explains the research problem in informal terms. Based on this, a fellow student on the course should be able to understand how your project relates to the rest of the course.
    • Theory (Materials and Methods). Describes the theory behind the application and/or methodology and cites books and scientific articles, where the theory can be found.
    • Simulation/Results. The method is applied to a simulated or real application. Codes can be included as appendices, if necessary.
    • Summary (Discussion and Conclusion). Summarizes the results and provides insight into the usability of the method. If applicable, it also discusses the good/bad sides of the approach.
    Codes can be returned as a separate archive or included into the appendix of the report if it is more convenient. 

    Use some word processing software (e.g. [pdf/xe/lua/...]LaTeX, preferably) to typeset your report. You can use some standard article or report template.