Stefan Geiss
Stochastic Processes and Applications
Time and Place : Monday
10-12 MaD 381
Wednesday 10-12 MaD 381
First lecture : 9/09/2002
Description:
Assume that You have a fair coin and
play the following game:
You and a second player have a starting
capital of about 100
Euro. The coin will be flipped and if
You have the tail, then
You get 1 Euro from the other player,
in the other case You have
to give her or him 1 Euro. One may ask
the following questions:
[Q1] What can You say about the
length of the game, if You assume
that the
game ends if You or Your partner has lost all the
money?
[Q2] What is the probability
that the game never ends?
If the coin is not fair, which may happen, one should ask:
[Q3] What is the probability that
You loose all Your money before
Your partner
looses all her or his money?
The course will answer these questions.
For example [Q1] is related
to the beautiful and surprising Law
of Iterated Logarithm.
Contents of the course:
1. Sums of independent random variables
(Zero-One Laws
and applications to random walks,
Strong law of
large numbers, Law of iterated logarithm)
2. Martingale theory
(Fundamental
inequalities according to Doob and Burkholder-Davis-Gundy,
Limit theorems
for uniformly integrable martingales)
3. Applications (Stochastic models)
Literature:
D. Williams: Probability with Martingales
(Cambridge)
A.N. Sirjaev: Probability (Springer)