Courses in Stochastics in Autumn 2008



MATS262/263 Todennäköisyysteoria 2 ja 3
Probability Theory 2 and 3

Time   Place
Wednesday 8-10 MaD 381
Thursday 8-10 MaD 380
First lecture 10th September 2008  

Computations for experiments often end up with sums of independent random variables and one is interested in their limit distributions.

First example, the Law of Large Numbers (LLN). Without this law statisticians and many people from Numerics could not live. But why do we have

\begin{displaymath}\frac{1}{N} \sum_{n=1}^N X_n \to {\rm I\! E}X_1 \end{displaymath}

for independent identically distributed integrable random variables? And before, in what sense does the convergence take place?

Second Example, the Law of Iterated Logarithm for a centered random walk, i.e. $\sigma^2 = {\rm I\! E}X_1^2<\infty$ and ${\rm I\! E}X_1=0$. A change of the scaling factor in the (LIL) yields the surprising

\begin{displaymath}{\rm I\! P}\left ( \limsup_n \frac{X_1+\cdots+X_N}{\psi(N)} =... ...with}\hspace{1 em} \psi(N) = \sqrt {2 \sigma^2 N \log \log N}.\end{displaymath}

How to explain the magic double logarithm?

The course will include both examples and provides the background concerning different modes of convergence of random variables.

Literature: A.N. Shirjaev, Probability (Springer)

Exercises and script:



MATA273 Rahoitusteorian stokastisia malleja 1
Models in Financial Mathematics 1

Time   Place
Monday 12-14 MaD 381
Tuesday 14-16 MaD 381
Tuesday 16-18 (Demo) MaD 380
First lecture 15th September 200827th October 2008  

Does Probability Theory help to make money at the stock exchange?

After introducing some basic probabilistic methods one can already start to investigate market models. We will compute "fair prices" for European and American options, find out whether "hedging" is always possible and, of course, answer the above question.

This course serves also as an introduction for MATA274 where we consider financial models in continuous time which are described by stochastic differential equations.

Literature:

Exercises:




MATA274 Rahoitusteorian stokastisia malleja 2
Models in Financial Mathematics 2

Time   Place
Monday 12-14 MaD 381
Tuesday 14-16 MaD 355
Tuesday 16-18 (Demo) MaD 380
First lecture 27th October 2008  
The course is dedicated to Financial Mathematics in continuous time. Therefore we will start with a short introduction to stochastic integration. We will discuss riskneutral pricing for continuous time market models and consider option pricing in the Black-Scholes model. Then we will shortly talk about the mathematical modelling of bonds and currency markets. At the end we will learn about the problem of incomplete markets and finally come to credit risk.

Literature:

Exercises: