Some Problems in the Theory of Risk (10 hours)

Monday 13        14.00-15.30
Tuesday 14    10.00-11.30    14.00-15.30
Wednesday 15    10.00-11.30    14.00-15.30

Summary
Risk theory is an active research topic in applied probability.
A risk process can be modeled by a compound Poisson model,
one sided Lvy process or compound renewal process. The ruin
is defined by a suitable transition of the risk process (e.g. below zero).
Such transitions define passage times and related probabilities
are called ruin functions (as e.g. ultimate ruin function,
finite horizon ruin function).

In the series of lectures we give an outline of the theory
and survey some recent trends. In particular the following points will be lectured:

I. Distributions in the theory of risk. In particular we outline
the theory of heavy-tailed distributions. We also mention other
classes like phase type and ME distributions.

II. Classical risk and Sparre-Andersen model. This will be an outline
of the basic ruin theory. We study here Cramer-Lundberg bounds and
Cramer-Lundberg approximations. In particular we discuss two cases
of interest: light and heavy tailed claim sizes.

III. Ruin probabilities for Levy processes. Compound risk process
is a special case of Lvy processes. We outline here some facts from
the theory of fluctuation of Lvy processes.

IV. Recent extensions. Recently new models were successfully studied
as e.g. Parisian ruin, model with dividends, model with taxes.
We outline here recent directions in risk theory.