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Stochastic methods in physics and biology, spring 2010

Lecturer

Stefan Geritz
Paolo Muratore-Ginanneschi

Scope

The aim of the course is to introduce the basic concepts of the theory of stochastic differential equations needed in applications (applied mathematics, physics and biology). In particular we will illustrate methods of qualitative, asymptotic and numerical analysis.

10 cu.

Type

Advanced studies.

Prerequisites

Lectures

Weeks 3-9 and 11-18, Monday 14-16 in room C124 and Friday 14-16 in room C123.

Easter holiday 1.-7.4.

First lecture: Friday, 22.01.2010

Lecture Notes

The lecture notes cover and sometimes integrate the material expounded in the lections. They also give bibliographic references for the same topics. For exercises please refer to the section below.

Lectures 1-10 Lectures 10-20
Lecture_01: Background of Probability Lecture_11: Kolmogorov-Chentsov theorem
Lecture_02 (rev 29.01): Bernoulli variables, Borel-Cantelli lemma Lecture_12: White-noise
Lecture_03: Limit theorems Lecture_13: Ito integral
Lecture_04: Martingales Lecture_14: Ito and Stratonovich calculus
Lecture_05: Numerical generation of random variables Lecture_15: SDE, existence and uniqueness
Lecture_06 (rev 12.02): Statistical tests Lecture_16: Kolmogorov pair and diffusion
Lecture_07: Fokker-Planck equation I Lecture_17: Exit time statistics
Lecture_08: Fokker-Planck equation II Lecture_18: Girsanov formula
Lecture_09: Brownian motion Lecture_19: Numerical integration of SDE's
Lecture_10: Karhunen-Loève representation Lecture_20: Population dynamics

Probabilistic interpretation of set theoretic concepts

Exams

There will be two options for taking the course. Passing an exam to be held after the end of the course, in
date to be agreed or working out a short project (recommended).

Bibliography

  1. L.C. Evans, "An Introduction to Stochastic Differential Equations", lecture notes.
  2. P. E. Kloeden, E. Platen, H. Schurz, "Numerical Solution Of Sde Through Computer Experiments", Springer (Universitext) (2003).
  3. N. Berglund, B. Gentz "Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach", Springer (Probability and its Applications) (2005).
  4. D.J. Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations" SIAM Review, Education Section, 43, 2001, 525-546. (Link to Higham's publications page.)

Book extended preview is available on line from google books

Registration

Did you forget to register? What to do.

Exercise groups

Group Day Time Place Instructor
1. Wednesday 16-18 C124  

Exercise set 1
Exercise set 2
Exercise set 3
Exercise set 4
Exercise set 5

Courseware

The codes below make use of the pgplot graphic library and of the GNU GSL
scientific library. You need to install these packages in order to use the codes as they are.

A c code to generate histograms and its initialization code