Stochastic differential equations

Timetable

30 hours lectures and 30 hours exercises.

Module aim (aims)

The aim of the course is to introduce the students to the
topic of differential equations depending on random noise.
The course will start from examples of such equations coming
from various fields of science such as physics or financial
mathematics. We will present pragmatic solution methods which work
in simple cases. To check if these solution methods are reliable,
we will delve into mathematical aspects of the subject.
We will reflect on the question how to model mathematically
random noise and how to integrate w.r.t. such noise. Then we
will prove the existence of solutions of equations which resist
simple solution methods.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Basic mathematical analysis and calculus. Some background in ordinary differential
equations, measure theory or probability will help, but is not mandatory.

Syllabus

- Outline of applications of stochastic differential equations (SDE).
From physics to financial mathematics.

- Preliminaries on ordinary differential equations.

- Pragmatic approach to differential equations with random noise:
Identification of new difficulties.

- Preliminaries on measure theory and probability.

- Stochastic processes, Wiener process, white noise.

- Ito integral.

- Existence and uniqueness of solutions of simple SDE.

- Applications revisited.

Reading list

B. Oksendal. Stochastic Differential Equations. An Introduction with Applications.
Springer, 2003.

J.M. Steele. Stochastic Calculus and Financial Applications. Springer, 2001.

T. Jahnke. A short introduction to stochastic differential equations.
Lecture notes, Karlsruhe Institute of Technology, 2023.

S. Saerkkae. Applied Stochastic Differential Equations. Lecture notes, 2012.