16.1 Introduction

Collective risk theory is concerned with random fluctuations of the total net assets - the capital of an insurance company. Consider a company which only writes ordinary insurance policies such as accident, disability, health and whole life. The policyholders pay premiums regularly and at certain random times make claims to the company. A policyholder's premium, the gross risk premium, is a positive amount composed of two components. The net risk premium is the component calculated to cover the payments of claims on the average, while the security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the average and also allows an accumulation of capital. So the risk process has the Cramér-Lundberg form:

$\displaystyle R(t) = u + ct - \sum_{k=1}^{N(t)} Y_k,
$

where $ u>0$ is the initial capital (in some cases interpreted as the initial risk reserve) of the company and the policyholders pay a gross risk premium of $ c>0$ per unit time, see also Chapter 14. The successive claims $ \{Y_k\}$ are assumed to form a sequence of i.i.d. random variables with mean $ \textrm{E}Y_k = \mu$ and claims occur at jumps of a point process $ N(t)$, $ t\geq 0$.

The ruin time $ T$ is defined as the first time the company has a negative capital, see Chapter 15. One of the key problems of collective risk theory concerns calculating the ultimate ruin probability $ \Psi = \textrm{P}(T < \infty \vert R(0)=u),$ i.e. the probability that the risk process ever becomes negative. On the other hand, an insurance company will typically be interested in the probability that ruin occurs before time $ t$, i.e. $ \Psi(t) =
\textrm{P}(T < t \vert R(0)=u)$. However, many of the results available in the literature are in the form of complicated analytic expressions (for a comprehensive treatment of the theory see e.g. Asmussen; 2000; Rolski et al.; 1999; Embrechts, Klüppelberg, and Mikosch; 1997). Hence, some authors have proposed to approximate the risk process by Brownian diffusion, see Iglehart (1969) and Schmidli (1994). The idea is to let the number of claims grow in a unit time interval and to make the claim sizes smaller in such a way that the risk process converges weakly to the diffusion.

In this chapter we present weak convergence theory applied to approximate the risk process by Brownian motion and $ \alpha $-stable Lévy motion. We investigate two different approximations. The first one assumes that the distribution of claim sizes belongs to the domain of attraction of the normal law, i.e. claims are small. In the second model we consider claim sizes belonging to the domain of attraction of the $ \alpha $-stable law ( $ 1<\alpha<2$), i.e. large claims. The latter approximation is particularly relevant whenever the claim experience allows for heavy-tailed distributions. As the empirical results presented in Chapter 13 show, at least for the catastrophic losses the assumption of heavy-tailed severities is statistically justified. While the classical theory of Brownian diffusion approximation requires short-tailed claims, this assumption can be dropped in our approach, hence allowing for extremal events. Furthermore, employing approximations of risk processes by Brownian motion and $ \alpha $-stable Lévy motion we obtain formulas for ruin probabilities in finite and infinite time horizons.