Classical insurance risk models rely on independent increments of the corresponding risk process. However, this assumption can be very restrictive in modeling natural events. For example, Müller and Pflug (2001) found a significant correlation of claims related to tornados in the USA. To cope with these observations we present here a risk model producing positively correlated claims. In the recent years such models have been extensively investigated by Gerber (1981,1982), Promislov (1991), Michna (1998), Nyrhinen (1998,1999a,b), Asmussen (1999), and Müller and Pflug (2001).
We consider a model where the time of the year influences claims. For example, seasonal weather fluctuations affect the size and quantity of damages in car accidents, intensive rains can cause abnormal damage to households. We assume the existence of good and bad periods for the insurance company in the sense of different expected values for claim sizes. This structure of good and bad periods produces a dependence of claims such that the resulting risk process can be approximated by the fractional Brownian motion with a linear drift. Explicit asymptotic formulas and numerical results can be derived for different levels of the dependence structure. As we will see the dependence of claims affects a crucial parameter for the risk exposure of the insurance company - the ruin probability.
Recall that the ruin time is defined as the first time the company has a negative capital.
One of the key problems of collective risk theory concerns calculating the ultimate ruin probability
i.e. the probability that the risk process ever becomes negative.
On the other hand, the insurance company will typically be interested in the probability
that ruin occurs before time
, that is
. In the next section we
present basic definitions and assumptions imposed on the model, and results which permit
to approximate the risk process by fractional Brownian motion. Section 17.3
deals with bounds and asymptotic formulas for ruin probabilities. The last section
is devoted to numerical results.