Numerical Integration of Stochastic Differential Equations

Goals

In this course we will introduce and study numerical integrators for stochastic differential equations. These numerical methods are important for many applications.

Mean-square  (dark gray) and asymptotic stability region (dark and light grays) of the explicit Milstein–Talay method.

Sample of an SPDE modelling the electric potential in a neuron at fixed time (left), as a space-time function (right).

Teacher

Prof. Assyr Abdulle

Assistant

Giacomo Garegnani will be present in his office, MA C2 614, on Monday from 13:00 to 14:00 to answer your questions.

Organization of the course

Course: Monday from 14h15 to 16h00 at CE1103
Exercise session : Monday from 16h15 to 18h00 at CE1103

Material

Some notions and notations of probability theory needed in the course: 

Exercises

Week Series Week Series Week Series
Series 1
Series 2
Series 3
 
Series 4
 
Series 5
 
Series 6
 
Series 7
 
Series 8
Series 9
 
Series 10
 
Series 11
Series 12

Bibliography

L. Arnold, “Stochastic Differential Equations, Theory and applications”.

L.C. Evans, “An Introduction to Stochastic Differential Equations”, AMS, 2013.

A. Einstein, “Investigations on the theory of the Brownian Movement”, Dover Publications, INC., 1956. 

D. Talay, “Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution”, Modélisation mathématique et analyse numérique, 1986. 

H-H. Kuo, “Introduction to Stochastic Integration”, Springer, 2005.

P.E. Kloeden, E. Platen, “Numerical Solution of Stochastic Differential Equations”, second edition, Springer, 1999.

G.N. Milstein, M.V. Tretyakov, “Stochastic Numerics for Mathematical Physics”, Springer, 2004.

Pour les rappels sur les probabilités on peut consulter:

R. C. Dalang et D. Conus, “Introduction à la théorie des probabilités”, 1ère édition, PPUR, 2014

R. Derrett, “Probability: Theory and Examples“, Cambridge University Press 2010.

A. Gut, “Probability: A Graduate Course”, 2nd édition, Springer, 2013.

Ch.E. Pfister, “Théorie des probabilités”, première édition, PPUR, 2014.