Another look at the Picard-Lefèvre formula for finite-time ruin probabilities

In the compound Poisson risk model, with discrete claim size distribution, Picard and Lefèvre [Picard, P., Lefèvre, C., 1997. The probability of ruin in finite-time with discrete claim size distribution. Scand. Actuarial J. 1, 58–69] derived a formula to compute the finite-horizon ruin probability. Here, some alternatives to this formula are proposed: exact recursive formulas which provide the distribution of time to ruin at once and a Seal-type formula which only involve probabilistic quantities. Depending on the comparison between the initial reserve and the total premium up to the finite horizon, their different interests are discussed by comparing their performances. The numerical stability of the formulas is then investigated, and disagreements in the existing literature about the detection of critical values are explained. convolutions for pseudo-compound distributions are introduced, and a theorem is stated in order to switch between formulas based on Appell polynomials and Seal-type formulas. This also provides a derivation of the Picard–Lefèvre formula from sample path properties.