# The surpluses immediately before and at ruin, and the amount of the claim causing ruin

## Details

Serval ID

serval:BIB_7458BEDD18F2

Type

**Article**: article from journal or magazin.

Collection

Publications

Institution

Title

The surpluses immediately before and at ruin, and the amount of the claim causing ruin

Journal

Insurance: Mathematics and Economics

Publication state

Published

Issued date

1988

Volume

7

Number

3

Pages

193-199

Abstract

In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u; x, y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap(x + y), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h(u; z) can be obtained from f(u; x, y). One finds, for example, that h(0; z) = azp(z).

Keywords

Ruin theory, Surpluses before and at ruin, Combination of exponential claim amount distributions

Create date

19/11/2007 10:33

Last modification date

20/08/2019 14:32