EEAEV  Vaihtuvasisältöinen opinto, Applied Stochastic Differential Equations, 01.07.202004.09.2020
This course space end date is set to 04.09.2020 Search Courses: EEAEV
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 handson 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 (simo.sarkka@aalto.fi) and Arno Solin (arno.solin@aalto.fi) in summertime 1.7.28.8.2020 and the exam is on week 31.8.4.9. The course assistant is Zheng Zhao (zheng.zhao@aalto.fi).
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 14:1516:00. The first contact session is 1.7. at 14:1516:00.
The course will be arranged as a flipped class type of course:
The coursebook is available online here: http://users.aalto.fi/~ssarkka/pub/sde_book.pdf and the example codes from the book here: https://github.com/AaltoML/SDE
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 14 on 31.7. and rounds 58 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 14:1516:00 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 14 on 31.7.
Rounds 58 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:
1.7.2020  Selfstudy task: ODE Basics (ch. 2)
Read Chapter 2 in the coursebook
ODE Basics Quiz is mandatory, deadline 5.7.2020, no exercises this time
Useful example code: https://github.com/AaltoML/SDE/blob/master/matlab/ch02_ex09_numerical_solution_of_odes.m
8.7.2020  Lecture 1: Pragmatic Introduction to Stochastic Differential Equations (ch. 3)
Lecture Slides
Lecture 1 Part 1: Stochastic differential equations
Lecture 1 Part 2: Stochastic processes in physics and engineering
Lecture 1 Part 3: Heuristic solutions of linear SDEs
Lecture 1 Part 4: Heuristic solutions of nonlinear SDEs, and Summary
Lecture 1 Quiz
Exercise round 1: 3.1, 3.2, 3.3
Useful example codes:
https://github.com/AaltoML/SDE/blob/master/matlab/ch03_ex07_time_varying_oscillator.m
https://github.com/AaltoML/SDE/blob/master/matlab/ch03_ex10_stochastic_spring_model.m
https://github.com/AaltoML/SDE/blob/master/matlab/ch03_fig02_two_views_of_brownian_motion.m
https://github.com/AaltoML/SDE/blob/master/matlab/ch03_fig10_white_noise.m
15.7.2020  Lecture 2: Itô Calculus and Stochastic Differential Equations (ch. 4)
Lecture Slides
Lecture 2 Part 1: Stochastic integral of Itô
Lecture 2 Part 2: Itô formula
Lecture 2 Part 3: Solutions of linear and nonlinear SDEs, Summary
Lecture 2 Quiz
Exercise round 2: 4.1, 4.2, 4.6
Useful example codes:
22.7.2020  Lecture 3: Probability Distributions and Statistics of SDEs (ch. 5)
Lecture Slides
Lecture 3 Part 1: FokkerPlanckKolmogorov Equation
Lecture 3 Part 2: Moments of SDEs
Lecture 3 Part 3: Statistics of linear SDEs, Markov, Markov Properties and Transition Densities SDEs, Summary
Lecture 3 Quiz
Exercise round 3: 5.1, 5.2, 5.4
29.7.2020  Selfstudy task: Linear stochastic differential equations (ch. 6)
Read Chapter 6 in the coursebook
Exercise round 4: 6.1, 6.3, 6.8
5.8.2020  Lecture 4: Numerical Solution of SDEs, Itô–Taylor Series, Gaussian Approximations (ch. 8 & 9)
Lecture Slides
Lecture 4 part 1: Introduction, Gaussian approximations of nonlinear SDEs
Lecture 4 part 2: ItôTaylor series of SDEs
Lecture 4 part 3: ItôTaylor series based numerical methods, Summary
Lecture 4 Quiz
Exercise round 5: 8.1, 8.2, 9.1
Useful example codes:
12.8.2020  Lecture 5: Stochastic Runge–Kutta Methods (ch. 8)
Lecture Slides
Lecture 5 part 1: Introduction, Runge–Kutta methods for ODEs
Lecture 5 part 2: Strong stochastic Runge–Kutta methods
Lecture 5 part 3: Weak stochastic Runge–Kutta methods, Summary
Lecture 5 Quiz
Exercise round 6: 8.3, 8.4, 8.5
Useful example codes:
19.8.2020  Lecture 6: Bayesian Inference in SDE Models (ch. 10)
Lecture Slides
Lecture 6: Bayesian Inference in SDE Models  Problem Formulation
Lecture 6: Discretetime Bayesian filtering and Smoothing
Lecture 6: Continuous/DiscreteTime and Continuous Bayesian Filtering and Smoothing
Lecture 6 Quiz
Exercise round 7: 10.2, 10.4, 10.5
Useful example codes:
26.8.2020  Selfstudy task: Parameter estimation in SDE models (ch. 11)
Read Chapter 11 in the coursebook
Exercise round 8: 11.1, 11.4, 11.9
Useful example codes:


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:
http://users.aalto.fi/~ssarkka/pub/sde_book.pdf
The examples codes of the book can be found here:
https://github.com/AaltoML/SDE
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 510 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 htransform
 Girsanov theorem
 Variational Bayes approximations of SDEs
 Series expansions of SDEs
 Small noise expansion approximations
 Numerical solution of FokkerPlanckKolmogorov equations
 Statespace methods for Gaussian processes
 Solution of PDEs with FeynmanKac
 Wiener/Feynman path integrals and SDEs
 Existence and uniqueness of SDEs
 Martingale representation theorem
 Taylor series expansions of moment equations
 Levyprocess driven SDEs
 Spatially distributed systems
 BlackScholes 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.
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.
