Beräkningsmetoder för stokastiska differentialekvationer, stodif08

Teachers

Anders Szepessy
Raul Tempone
Håkon Hoel

Stochastic molecular dynamics of liquid-solid phase transition

Course material

- lecture notes (chapters 1, 9, 10 and 11 updated)
- Raul's slides
- Raul's slides
- Raul's slides
- Raul's slides

Lecture and Homework Schedule

Starting Tuesday March 25, 2008, at 13.15-15 in room V22.
Schedule for lectures and homework
The lecture March 31 is cancelled and moved to a later date.
The lectures on April 8 and April 22 are moved to April 9 and April 23 at 13-15 in room 4523 (CSC KTH).

The course focus on the following real-world problems and mathematical and numerical methods to solve them. In each application we study relevant mathematical and numerical methods to solve the problem. This includes methods and theory for ordinary, partial and stochastic differential equations, and optimal control, treating e.g. weak and strong approximation, Monte Carlo methods, variance reduction, large deviations for rare events.

 
Week	 Problem                        Subject

13,14	                                Stochastic DE, Ito-calculus, 
         stocks with noise              Euler method, 
         molecular dynamics             weak and strong convergence


15,16    Option price                   The Feynman-Kac formula,
         American options               Monte-Carlo Methods, variance reduction,
                                        finite difference methods.

17,18    Optimal hedging                Calculus of Variations, Optimal Control
         Reaction rates                 dynamical programming, Hamilton-Jacobi
                                        equations, large deviations and rare events

19,20    Implied volatility             Inverse problems, optimal control
         
 
20       presentations:
         geophysical flow,             
         turbulent diffusion            Convection-diffusion equations, wavelets
         Ground water flow              correlated noise
         material science
         networks
         reactions

                                   

Homework, Computer Lab's , Presentations and Examination

The Examination consists of three parts: Homework problems, oral presentations and a written exam. The homework problems will be available here on the course www-page. The homework and the presentations are carried out by groups of students. Each group hand in a report on each assignment.

Homework 1 on Ito integrals, due week 14.
Homework 2 on Ito and Stratonovich, due week 15.
Homework 3 on Feynman-Kac and Options, due week 16.
Homework 4 on Monte Carlo for Options, due week 17.
Homework 5 on Dynamic Programming, due week 19.
Alternative 5 on Reaction rates, due week 19.

Suggestions for presentation of the computational problems on week 19.
Suggestions for the homework. (Swedish)

Handout materials for the lectures

An introductory article on numerical simulation of SDE:s by Desmond J. Higham, available from the KTH library: An algorithmic introduction to numerical simulation of stochastic differential equations, by D.J. Higham, SIAM Review, Education Section, 43, 2001, 525-546.
The source files for the examples in the article are available here.

Additional source files by Jesper Carlsson.
Example matlab code for example 5.13

Participants and Result

Here is the list of participants and result of this course.

Exam Paper

A substantial part of the exam will be based on a list of questions given
for 2008 here
The maximal score will be 60, and to pass the course you must obtain a total score, homework included, of approximately 60. The homework and the presentation gives maximal 40 credits together, with maximal 5 credits for each homework 1-4 and maximal 10 credits for homework 5 and for the final presentation.

Exam

The exam will be in room 4523 (CSC) at 13.15 May 26, 13.15 May 30, and 10.15 June 10 2008 (Choose one of the three possible dates).

Student's comments

Here is a course evaluation form

Sample paths of solutions to stochastic differential equation and its probabability density