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Department of Information Technology

Numerical methods in stochastic modeling and simulations

PhD course (7.5hp)

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The course covers (1) a brief introduction to the theory of stochastic differential equations (SDEs) and a slightly more involved discussion on numerical solutions thereof, (2) Markov Chain Monte Carlo methods and in particular continuous-time Markov chains and discrete state space models of the Ising type, and (3) parameter inference in SDEs. Notably, some methods studied in (1) and (2) are combined in the problems discussed in (3).

The course will be given for the second time during the spring 2020 (period 3). Take a look at the evaluation from last time the course was given.

This is mainly a project-based course, with supporting lectures scheduled in between the mandatory seminars for which you are required to prepare exercises or small projects. The projects are open to incorporate lots of ideas of your own, please do!



Moment Time and place To prepare
Introductory lecture 2020-01-16, 13:15--15:00, ITC Come as you are
P1: Seminar 1 2020-01-31, 13:15--15:00, ITC Exercises
P2: Seminar 1 2020-02-14, 13:15--15:00, ITC Miniproject 1
P3: Seminar 1 2020-02-28, 13:15--15:00, ITC Miniproject 2
End 2020-03-20 Deadline Miniproject 3

To prepare means that you should submit a concise and formatted report (not handwritten) before the scheduled event. If the report happens to be in draft version, no worries, you then submit a final version before the next scheduled event after possibly receiving some feedback on your draft. The more prepared you are, the more effective and useful will the seminar be! Do submit before each seminar!

Come as you are means that the lecture is not mandatory, do pop in as you like. The purpose of these events (3 scheduled in total) is to support the Miniproject. I will prepare skeleton solutions, details to discuss, hints and suggestions... The more you have looked into the Miniproject, the more useful will these events be!

To pass the course you should submit all assignments and participate actively on all Seminars (4 scheduled in total). If you miss one Seminar event, an extra assignment will need to be submitted. Try very hard not to miss more than one seminar event!

Description of the course

The course is divided into three parts. All parts end with a "miniproject" to be submitted in the form of a written report.

Introductory Lecture

  • Stochastic modeling; complex dynamical systems; uncertainty propagation; stochastic modeling and numerical methods
  • What is in this course and what is not
  • Set-up and information concerning the course
  • Effective summary of basic probability theory; stochastic processes; stability and convergence

The first lecture is not mandatory. If you cannot come to the first lecture but wishes to take the course, be sure to let me know in order to receive information.

Part 1

  • SDE: basic theory specifically aiming at introducing those context used in Numerical analysis, like existence/uniqueness and tools and results in obtaining a priori bounds (§1-5 in Øksendal´s book).
  • Numerical methods for SDEs: methods for discretization, strong/weak convergence, (SDEs with jumps), exact simulation of SDEs (part of the material is found in Part IV-VI of Kloeden and Platen´s book).
  • Part 1 will be covered in 2x2 hour seminars.

To prepare in Part 1:

  • Seminar 1: Exercises and reading in §1-5 of Øksendal´s book "Stochastic Differential Equations", Springer 2003, 6th edition:
    • §1: read!
    • §2.1: stochastic process, §2.2: Brownian motion. Exercises: 2.4, 2.8.
    • §3.1: Itô integral and isometry, §3.2: properties, (§3.3: Stratonovich interpretation). Exercises: 3.1, 3.5, 3.13.
    • §4.1+(4.2): Itô formula, §4.3: Itô representation. Exercises: 4.1, 4.2, 4.7.
    • §5.1: Wiener SDEs, §5.2: Existence and Uniqueness (important!), (§5.3: Weak and Strong solutions). Exercises: 5.1, one of 5.5 or 5.7, 5.10, one of 5.12 or 5.15, 5.17.
    • Note: you can get substantial help at the end of the book.
    • Submit your (draft) solution to exercises no later than 2020-01-31 @ 13.00!
  • Seminar 2: Miniproject:

Part 2

  • Monte Carlo methods for Ising-type models, (variance reduction), (quasi-Monte Carlo and randomized quasi-Monte Carlo), and continuous-time Markov chains. Material from the book by Newman and Barkema will be used here
  • Continuous-time Markov chains as a limit, (time discretization thereof)
  • (Piecewise deterministic Markov processes and multiscale modeling)
  • Part 2 is scheduled as a 2 hour seminar.

To prepare in Part 2:

Part 3

  • Maximum-Likelihood/Bayesian frameworks for estimation of parameters
  • Parameter estimation for SDEs (likelihood-based)
  • Practical use of Markov chain Monte Carlo (Metropolis algorithm)
  • Part 3 is scheduled as a 2 hour seminar.

To prepare in Part 3:

Updated  2020-02-28 09:28:33 by Stefan Engblom.