**Seminar on Stohastics**

**PROGRAM**

Matematicki institut SANU

Kneza Mihaila 35/1

PLAN RADA SEMINARA ZA OKTOBAR 2002.

Ovog meseca gost iz Bugarske profesor Elisaveta Panceva odrzace kratak kurs iz teorije rizika na nasem seminaru. Molimo da obratite paznju na to da u petak 18. oktobra predavanje pocinje u 11 h. Prvi deo tog predavanja bice zajednicko sa Odeljenjem za matematiku.

Cetvrtak, 17. oktobar u 12 h.

* dr Elisaveta Panceva: *

1.Classical model of insurance risk.

2. Probability of ruin when the claim times form a renewal process.

Petak, 18. oktobar u 11 h.

3. Ruin theory for heavy-tailed claims.

4. Sample paths properties of the risk process.

5. VaR - a measure of extreme risk.

The aim of this mini-course is to introduce the participants in the basic models of insurance risk and their theoretical background. It supposes some basic knowledge on Probability and Statistics and Random Processes.

The classical insurance model is described by three random processes: the claim size process $(X_{k}),k\geq 1$, the arrival time process $ (T_{k}),k\geq 1$, and the counting process $N(t)$ equal to the number of claims in the interval $[0,t]$.

In the simplest model, the Cramer-Lundberg model, the claims $X_{k}$ are supposed iid and the interarrival times $\tau _{k}=T_{k}-T_{k-1}$ are independent and exponentially distributed. Then the counting process $N(t)$ is Poisson. In the case of iid interarrival times with arbitrary distribution, the counting process is a renewal process.

The risk process of the underlying portfolio is defined as \[ R(t)=u+ct-\sum_{k=1}^{N(t)}X_{k} \] Here $u$ is the initial capital and $c>0$ denotes the premium income rate. As a measure of risk we take the probability of ruin $P(R(t)<0,t>0)$. Besides the Poisson and renewal models, we also consider the case of heavy-tailed claims and discuss how and when the ruin occurs. Finally, we calculate the Value at Risk (VaR) as a measure of extreme risk. The main stochastic tools we use are some elements of renewal theory, random walk, regularly varying functions and statistical estimation of tails and quantiles.

Slobodanka Jankovic i Svetlana Jankovic

rukovodioci seminara