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

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).