Numerical methods in stochastic modeling and simulations

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 first time during the spring 2016 (period 4).

Contact: Stefan Engblom.

Moment	Time and place	To prepare
Introductory lecture	Thu 2016-04-07, 13:15--15:00 2244, ITC	Come as you are!
P1: Seminar 1	Thu 2016-04-14, 13:15--15:00 2345, ITC	Exercises
P1: Seminar 2	Thu 2016-04-21, 10:15--12:00 1113, ITC	Miniproject
P2: Seminar 1	Wed 2016-05-04, 13:15--15:00 2345, ITC	Miniproject
P3: Seminar 1	Tue 2016-06-14, 13:15--16:00 2345, ITC	Presentation+Miniproject

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!

To pass the course you should submit all assignments and participate actively on all occasions (except for the first introductory lecture). If you miss one event, an extra assignment will need to be submitted. Try very hard not to miss more than one event!

If you find (legal) links to books, useful articles, external resources, Wikipedia-articles... let me know and I will post them here!

Introducing Monte Carlo Methods with R

An Introduction to Stochastic Calculus with Applications to Finance

Computer intensive methods in statistics (link will break after the course is finished!)

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, (transformation methods), (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 2016-04-13 @ 13.00!
• Seminar 2: Miniproject:
  • Convergence of numerical methods for SDEs
  • Submit your (draft) report no later than 2016-04-20 @ 13.00!

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:

• Seminar 1: Miniproject:
  • Monte Carlo simulation of an Ising model
  • Submit your (draft) report no later than 2016-05-04 @ 13.00!

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:

• Seminar 1: Miniproject:
  • 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 2016-06-13 @ 13.00!
  • This is also the deadline for the final version of all previous miniprojects and exercises.
  • For this seminar, please communicate with me what extended task you like to perform and present. Presentation time 15 minutes, excluding questions and a discussion.