The Time of Ruin in the Classical Poisson Risk Model
Distribution and moments involving the time of ruin is the subject matter of this chapter. Moments of the time of ruin are considered in Sect. 6.1. For the distribution of the time of ruin, one approach involves analytic inversion of the Laplace transform of the time of ruin, a special Gerber-Shiu function. An alternative argument, introduced to the actuarial community by Seal in a series of publications, involves the method of infinitesimals seems to result in somewhat simpler formulas, however. In Sect. 6.2 this infinitesimal approach results in a partial integrodifferential equation for the finite time ruin probabilities. Tractable computational formulas for the finite time ruin probabilities in the case of mixed Erlang claims are derived in Sect. 6.3. The argument of Sect. 6.2 is then applied in Sect. 6.4 to obtain the joint distribution function of the time of ruin and the deficit. A combination of the Gerber-Shiu and Seal methodology is used to derive a somewhat different formula for the density of the time of ruin in Sect. 6.5.