EEA-EV - Vaihtuvasisältöinen opinto, Applied Stochastic Differential Equations, 01.07.2020-04.09.2020
This course space end date is set to 04.09.2020 Search Courses: EEA-EV
Topic outline
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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 (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 14:15-16:00. 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: 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 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.
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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:
1.7.2020 - Self-study 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 non-linear 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 non-linear 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: Fokker-Planck-Kolmogorov 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 - Self-study 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 non-linear SDEs
Lecture 4 part 2: Itô-Taylor series of SDEs
Lecture 4 part 3: Itô-Taylor 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: Discrete-time Bayesian filtering and Smoothing
Lecture 6: Continuous/Discrete-Time and Continuous Bayesian Filtering and Smoothing
Lecture 6 Quiz
Exercise round 7: 10.2, 10.4, 10.5
Useful example codes:
26.8.2020 - Self-study 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:
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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.
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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.
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.
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