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!
If you find (legal) links to books, useful articles, external resources, Wikipedia-articles... let me know and I will post them here!
Part 1, Oksendal: Stochastic Differential Equations
Part 2, Newman, Barkena: Monte Carlo methods in statistical physics
Part 3, Zwanzig, Mahjani: Computer Intensive Methods in Statistics
Robert, Casella: Introducing Monte Carlo Methods with R
Calin: An Introduction to Stochastic Calculus with Applications to Finance
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.
- 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.
- 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:
- Miniproject 1: Convergence of numerical methods for SDEs
- Submit your (draft) report no later than 2020-02-14 @ 13.00!
- 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:
- Seminar 1: Miniproject:
- Miniproject 2: Monte Carlo simulation of an Ising model
- Submit your (draft) report no later than 2020-02-28 @ 13.00!
- 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:
- Seminar 1: Miniproject:
- Miniproject 3: Estimation of SDE parameters through Markov chain Monte Carlo
- GBM data challenge (.mat-file). Variable 'X' holds the trajectories (each column is one trajectory), variable 'tspan' holds the sampling points in time.
- Submit your (draft) report no later than 2020-03-20 @ 13.00!
- This is also the deadline for the final version of all previous miniprojects and exercises.