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‎LICENSE

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Copyright Zheng Zhao and all lecturers (add their names here) - All rights reserved
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The course materials are proprietary, however, the authors grant you the right to freely download and use the materials.
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The codes in folder `./***` are distributed under the MIT license.
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Copyright Zheng Zhao and all the lecturers (add their names here) - All rights reserved

‎README.md

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# A computational introduction to stochastic differential equations
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To be updated.
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This course aims to develop a computational view of stochastic differential equations (SDEs) for students that have an applied or engineering background, e.g., machine learning, signal processing, electrical engineering, control, and statistics.
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# Course arrangements
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# Prerequisites
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1. Linear algebras
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2. Real analysis (not essential)
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3. Probability theory
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4. Ordinary differential equations
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# How to register
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Please fill in the Google form https://forms.gle/mC7tLBUnPdEUL4XeA to register the course.
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# Essential lectures (6 credits)
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1. **Introduction to the course**. <br>
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17 Oct, 2022. Room 4005 Ångström.
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2. **Stochastic differential/integral equations**. <br>
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21 Oct, 2022. Room 101132 Ångström.
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3. **Numerical solution to stochastic differential equation**. <br>
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24 Oct, 2022. Room 101127 Ångström.
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4. **Statistical properties of SDE solutions**. <br>
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28 Oct, 2022. Room 101142 Ångström.
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5. **Linear SDEs and Gaussian processes**. <br>
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31 Oct, 2022. Room 101146 Ångström.
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6. **Exercise 1**. <br>
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2 Nov, 2022. Room 101127 Ångström.
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7. **Filtering and smoothing problems I (i.e., regression with SDE prior)**. <br>
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4 Nov, 2022. Room 101127 Ångström.
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8. **Filtering and smoothing problems II (i.e., regression with SDE prior)**. <br>
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7 Nov, 2022. Room 101127 Ångström.
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9. **Exercise 2**. <br>
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9 Nov, 2022. Room 101127 Ångström.
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10. **SDE system identification**. <br>
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11 Nov, 2022. Room 101142 Ångström. <br>
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Lecturer: [Mohamed Abdalmoaty](https://people.kth.se/~abda/)
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11. **Exercise 3**. <br>
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18 Nov, 2022. Room 101127 Ångström.
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12. **Student project work presentation**. <br>
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16 Dec, 2022. Room 101142 Ångström.
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Time is 13:15 - 17:00 for all the lectures.
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# Seminar lectures (9 credits)
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By attending (not necessarily all) the seminar courses and complete their writing assigments/exericses, you get upgrade to 9 credits.
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1. **Continuous-time filtering**. <br>
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This lecture is concerned with continuous-time filtering, for example, Zakai equation, Kushner equation, and projection filtering. <br>
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Date: 14 Nov, 2022. Room 101127 Ångström <br>
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Lecturer: [Muhammad Fuady Emzir](https://scholar.google.com/citations?user=nfBRAHAAAAAJ&hl=en) (KFUPM)
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2. **SDEs and Markov chain Monte Carlo**. <br>
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In this lecture, we present a general recipe for constructing Markov chain Monte Carlo (MCMC) samplers, including stochastic gradient (SG) versions, from stochastic continuous dynamics (SDEs). We also explore the connections between SG-MCMC and stochastic optimization methods via simple annealing techniques. Recommended readings: 1) A Complete Recipe for Stochastic Gradient MCMC. 2) Bridging the gap between stochastic gradient MCMC and stochastic optimization. <br>
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Date: 21 Nov, 2022. Room 101127 Ångström <br>
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Lecturer: [Cagatay Yildiz](https://cagatayyildiz.github.io/) (University of Tübingen)
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3. **Probabilistic numerics for ordinary differential equations**. <br>
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Probabilistic numerical methods aim to explicitly represent the numerical uncertainty that results from limited computational resources. In this lecture, we present a class of probabilistic numerical solvers for ODEs which pose the numerical solution of an ODE as a Gauss--Markov regression problem. The resulting methods, called "ODE filters", efficiently compute posterior distributions over ODE solutions with methods from Bayesian filtering and smoothing. <br>
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Date: 25 Nov, 2022. Room 101127 Ångström <br>
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Lecturer: [Nathanael Bosch](https://nathanaelbosch.github.io/) (University of Tübingen)
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4. **Applications of SDEs in statistical signal processing**. <br>
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In this lecture we present a few applications that use SDEs to solve signal processing problems. These include, for example, using SDEs to construct non-stationary processes to model complicated signals, and using SDEs to estiamte the spectrogram of signals with uncertainty. <br>
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Date: 28 Nov, 2022. Room 101127 Ångström
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5. TBD. <br>
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2 Dec, 2022. Room 101127 Ångström. You are very welcome to contact me if you would like to give a guest lecture.
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6. **Constructions of Wiener processes and stochastic integrals**. <br>
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This lecture explains the constructions of Brownian motion and Ito integrals.
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Date: 5 Dec, 2022. Room 101127 Ångström
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Time is 13:15 - 17:00 for all the lectures.
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Note that the dates for the seminar courses are not fixed. They are subject to change depending on the schedule of the lecturers.
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# Course arrangement
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The course consists of lectures, exercises, and project work. Specifically, in each week, there would be one/two lectures (45 + 45 mins) and an exercise session (60 mins). The students shall present and discuss their exercise solutions during the exercise session.
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Total credit is 6 or 9.
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In order to get 6 credits, you need to
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- actively participate all the essential lectures,
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- pass the three exercise assignments,
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- present the project work. Depending on the number of students, you may do the project work in group.
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If you would like to get 9 credits, you need to fullfill the requirements for the 6 credits as stated above, and in addition,
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- actively participate all the seminar lectures,
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- Select five from all the seminar lectures, then pass the exercises of the selected, or do a writing assignment if the lecture has no exercise. (We will define the writing assignment later).
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The course grade is based on pass/fail.
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# Project work
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To be added.
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# Reading materials
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This course is mainly based on the following textbooks.
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- Hui-Hsiung Kuo. Introduction to stochastic integration. Universitext. Springer, 2006.
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- Simo Särkkä and Arno Solin. Applied stochastic differential equations. Cambridge University Press, 2019.
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- Ioannis Karatzas and Steve E. Shreve. Brownian motion and stochastic calculus. Springer, 2nd edition, 1991.
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# Course history
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- Oct - Dec, 2022, Uppsala Universitet, FTN0332 TN22H006.
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# Contact
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Zheng Zhao, Uppsala University.
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[email protected]
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https://zz.zabemon.com

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