# Seminar on Stochastic Geometry and its applications

## Seminar Supervisor

Prof. Dr. Evgeny Spodarev

Prof. Dr. Volker Schmidt

## Important Information

There will be no seminar on the first Monday of the upcomming summer term (April, 13th). But there will be a short information meeting on the first monday of the upcomming summer term. If you should be interested in participating in the seminar, do not hesitate to come to my office (Room 141, Heho 18) on Monday, April 13th, between 02:15pm and 04:00pm.

## Seminar Advisor

Alexander Nerlich

In case you would like to participate in the seminar, do not hesitate to contact Alexander Nerlich.

## Date and Place

Time: Monday, 02pm - 04pm, Place: Raum 2002, in O28

## Prerequisites

The audience is supposed to be familiar with basic probability and measure theory. Considering the list of talks one may assumes that knowledge about Ito-Integrals is a prerequisite for this seminar, but we would like to ensure the reader, that all knowledges on Ito Integrals will be taught during the seminar.

## Intended Audience

Bachelor and Master Students of any mathematical study course.

The "(B)" in the list of talks, means that this talk is supposed to be given by a Bachelor's Student, the (M) indicates that the talk is supposed to be given by a Master's Student and consequently (B/M) indicates that the the talk is supposed to be given by either a Bachelor's or a Master's Student.

## Content

The aim of this seminar is to give an introduction to stochastic differential

equations (SDEs) and its application. The seminar will be structured as

follows:

1. At first we will introduce Martingales and Brownian Motions.

2. Then we will give an introduction to the so called Ito Integral.

3. Then we will use this introduction to discuss SDEs.

4. Finally we will discuss some applications of SDEs.

Most, but not all, applications will be related to finance. The applications

related to finance will include for example markets, arbitrage and option

pricing. Hereby it is common to model markets by SDEs, which are based on

Ito Integrals.

To get an imagination: Here is a possible trajectory of the (probably most

popular) Ito Integral.

(The picture shows the integral of the Browninan Motion integrated with respect to itself.)

A (not to finance related) application which we will obtain, will be the following

recurrence result about Brownian Motions:

Imagine there is somewhere in the 2-dimensional space a circle and a man

walks completely arbitrary on this plane. (This means he does one step in an

arbitrary direction and then he does another step in an arbitrary direction,

independent of the step he did before, and so on...) The question which

arises is: How high is the probability that the man hits the circle? From a

theoretical point of view (and under certain assumptions) one could model

this situation by saying that the movement of the man is the trajectory of a (2

dimensional) Brownian motion. And we will use this modeling approach to

show that the man will hit the circle with probability one. Furthermore we will

show that this probability is strictly less than one if one considers the same

scenario with a ball in three dimensions and a 3 dimensional Brownian

motion. (One can imagine the 3 dimensional brownian motion (under certain

assumptions) as a randomly moving bird.) In mathematics this result is

known under the following phrase: “A drunk man always finds his way home,

but a drunk bird can get lost forever.”

**Criterias to pass the Seminar**

Each student is supposed to give a talk and to attend the seminar on a regular basis. Those who give a (good) talk and attend the seminar regularely will pass the seminar.

## List of Talks

20th of April: Bennet Ströh The Brownian Motion (B) Vortrag

27th of April: Stochastic Processes and Martingales (B) Vortrag

04th of May: The construction of the Ito Integral (B/M) Vortrag

11th of May: Some basic properties of the Ito Integral (B/M) Vortrag

18th of May: The Ito Formula, its proof and its applications (B/M) Vortrag

1st of June: The m-dimensional Ito Integral and the m-dimensional Ito Formula (M) Vortrag

8th of June: An introduction to SDEs with some examples (M) Vortrag

15th of June: The uniqueness of strong solutions (M) Vortrag

22nd of June: The Markov property (M) Vortrag

06th of July: A drunk man always finds his way home, but a drunk bird can get lost

forever (M)

13th of July: Markets, Portfolios and Arbitrage (M) Vortrag

# Kontakt

**Seminar Advisor**

Alexander Nerlich (E-Mail)

Office: 141, Heho 18, Office hours by appointment.

Tel.: 0731 50 23537