### Stochastische Differentialgleichungen (Winter Semester 2018/19)

• Lecturer:
• Classes: Lecture (0105500), Problem class (0105510)
• Weekly hours: 4+2

PLEASE NOTE: There are are no lectures and tutorials on the following dates

• November 22: no tutorial - alternative date: December 20, 5:30 - 7pm in room 3.069
• November 23: no lecture - alternative date: 2019
• November 29: no tutorial - alternative date: February 7, 5:30 - 7pm in room 3.069
Schedule
Lecture: Tuesday 11:30-13:00 SR 3.069
Friday 9:45-11:15 SR 3.069
Problem class: Thursday 15:45-17:15 SR 3.069
Lecturers
Lecturer Prof. Dr. Lutz Weis
Office hours:
Room 2.047 Kollegiengebäude Mathematik (20.30)
Email: lutz.weis@kit.edu
Problem classes Dr. Markus Antoni
Office hours: by appointment
Room 2.044 Kollegiengebäude Mathematik (20.30)
Email: markus.antoni@kit.edu

In the mathematical approach to models appearing e.g. in nature sciences, engineering or finance which are described by an ordinary differential equation

we sometimes have to take random perturbations of the system into account. These 'errors' may originate from measurement errors, random environmental impacts, or incomplete information of the system's behaviour. These models can be described by stochastic differential equations of the form

where is a Brownian motion and is a 'white noise'.

The purpose of this lecture is to study solutions of such stochastic differential equations. In order to do that we will start with a short recap of basic stochastic concepts (e.g. CDFs, independent random variables) and continue with an introduction to stochastic analysis (stochastic integrals, Ito formula, martingales, stopping times). With these results at hand we will prove fundamental existence, uniqueness, and stability results for stochastic differential equations as well as properties of their solutions (Markov property, path regularity).

This theory will be illustrated by examples appearing in mathematical finance, physics, biology, and engineering.

Prerequisites: Integration theory, basic stochastic concepts, Hilbert spaces

### Exercise sheets

 Exercise sheet 1 Exercise 3 d)-e) Exercise sheet 2 Solutions 2 Exercise sheet 3 Exercise sheet 4 Exercise 12 Exercise sheet 5 Solutions 5 Exercise sheet 6 Solutions 6 Exercise sheet 7 Exercises 19 c) and 21 c) Exercise sheet 8 Exercise sheet 9 Exercise sheet 10 Exercise 30 b) Exercise sheet 11 Exercise sheet 12 Exercise sheet 13

### Sample exam

The sample exam will take place on Thursday, February 7, 2019, 3:45 - 7pm, in room 3.069 of the Kollegiengebäude Mathematik (20.30).

### Exam

The final exam will take place on Thursday, February 21, 2019, 11am - 1pm, in room 1.067 of the Kollegiengebäude Mathematik (20.30).

The preliminary results of the exam can be found on the blue board next to room 2.041 or online in CAS.

The post-exam review will take place on Thursday, March 14, 2019 at 1pm in room 2.070 of the Kollegiengebäude Mathematik (20.30).

### Literature

• J.M. Steele: Stochastic Calculus and Financial Applications, Springer, 2001
• L.C. Evans: An Introduction to Stochastic Differential Equations, AMS, 2013
• X. Mao: Stochastic Differential Equations and Applications, Woodhead Publishing, 2008