Risk theory

Collective risk theory deals with stochastic models of the risk business of an insurance company. In such a model the occurrence of the claims is described by a point process and the amounts of money to be paid by the company at each claim by a sequence of random variables $X_1,X_2,\dots$. The company receives a certain amount of premium to cover its liability. The company is furthermore assumed to have a certain initial capital $u$ at its disposal. One important problem in risk theory is to investigate the ruin probability $\Psi(u)$, i.e., the probability that the risk business ever becomes negative.

The classical risk model is defined as follows:

i) the stochastic point process is a Poisson process with intensity $\lambda$;

ii) the costs of the claims are described by independent and identically distributed random variables (cf. Random variable) $X_1,X_2,\dots$, having the common distribution function $F$, with $F(0)=0$, and mean value $\mu$; it is assumed that $h(r)=\int_0^\infty(e^{rz}-1)dF(z)<\infty$ for some $r>0$;

iii) the point process and the random variables are independent;

iv) The premiums are described by a constant (and deterministic) rate $c$ of income.

Let the relative safety loading $\rho$ be defined by

$$\rho=\frac{c-\lambda\mu}{\lambda\mu}$$

and let the Lundberg exponent $R$ be the positive solution of

$$h(r)=\frac{cr}{\lambda}.$$

The following basic results for the classical risk model go back to the pioneering works [a8] and [a3]; here,

$$\Psi(0)=\frac{1}{1+\rho};$$

when the claim costs are exponentially distributed:

$$\Psi(u)=\frac{1}{1+\rho}e^{-(\rho u)/\mu(1+\rho)};$$

the Cramér–Lundberg approximation:

$$\lim_{u\to\infty}e^{Ru}\Psi(u)=\frac{\rho\mu}{h'(R)-c/\lambda};$$

the Lundberg inequality:

$$\Psi(u)\leq e^{-Ru}.$$

The classical risk model can be generalized in many ways.

A) The premiums may depend on the result of the risk business. It is natural to let the safety loading at a time $t$ be "small" if the risk business, at that time, attains a large value and vice versa.

B) Inflation and interest may be included in the model.

C) The occurrence of the claims may be described by a more general point process than the Poisson process.

[a4] and [a5] focus mainly on A) and B). In [a1] and [a9] generalizations of i) to renewal processes (cf. also Renewal theory) are discussed. The monographs [a2] and [a7] treat, among others, the case where the claims occur according to a Cox process. Large claims, where the assumption $h(r)<\infty$ for some $r>0$ does not hold, are treated in [a6].

References

[a1]	E. Sparre Andersen, "On the collective theory of risk in the case of contagion between the claims" , Trans. XVth Internat. Congress of Actuaries , II , New York (1957) pp. 219–229
[a2]	S. Asmussen, "Ruin probability" , World Sci. (to appear)
[a3]	H. Cramér, "On the mathematical theory of risk" Skandia Jubilee Volume (1930)
[a4]	A. Dassios, P. Embrechts, "Martingales and insurance risk" Commun. Statist. - Stochastic models , 5 (1989) pp. 181–217
[a5]	F. Delbaen, J. Haezendonck, "Classical risk theory in an economic environment" Insurance: Mathematics and Economics , 6 (1987) pp. 85–116
[a6]	P. Embrechts, N. Veraverbeke, "Estimates for the probability of ruin with special emphasis on the possibility of large claims" Insurance: Mathematics and Economics , 1 (1982) pp. 55–72
[a7]	J. Grandell, "Aspects of risk theory" , Springer (1991)
[a8]	F. Lundberg, "Försäkringsteknisk Riskutjämning" , F. Englunds (1926) (In Swedish)
[a9]	O. Thorin, "Probabilities of ruin" Scand. Actuarial J. (1982) pp. 65–102