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

Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes

“…For c > 0, x > 0 and n ∈ N \ {0}, we have This is a direct adaptation of results of (Rullière and Loisel (2004)). It thus suffices to study the cumulated claim amount process at a finite set of inventory dates.…”

“…ψ(u, t) = P [∃s ∈ [0, t], R s < 0 | R 0 = u] , u 0, t > 0, and let ϕ(u, t) = 1 − ψ (u, t) be the probability of non-ruin within time t with initial reserve u. Algorithms to compute or approximate ψ(u, t) have been proposed, among others, by (Asmussen, Avram and Usabel (2002)) by the means of Erlangization, by (Picard and Lefèvre(1997)) by the means of Appell polynomials, and by (Rullière and Loisel (2004)) by the means of a Seal-type argument based on the ballot Lemma. An important feature of Solvency II is that estimation risk should be controlled, particularly if internal models are used.…”

On finite-time ruin probabilities for classical risk models

“…, S t+u evaluated at different points between 0 and u + t−1. Comparing the efficacy of (2.18) and (2.33) is rather delicate (see, e.g., Rullière and Loisel [28]). Some numerical experiments show that (2.33) becomes especially fast when u is small compared to t.…”

“…In Picard and Lefèvre [24] and De Vylder and Goovaerts [10], the first step consists in stipulating for a total claim amount until some time t and conditioning on the last claim instant and amount before t. This yields an integral equation which is then solved by the former authors using the generalized Appell polynomials (in an extended framework), and by the latter authors using the convolution products and the Laplace transforms. On another hand, Rullière and Loisel [28] showed that the P.L. formula can be linked to the well-known ballot theorem and a Seal-type formula (see, e.g., Seal [29] and Gerber [12]).…”

“…Such a case is important because in practice a et al [17]. A refined discussion of its merits in comparison with other available formulas is given in Rullière and Loisel [28]. Besides, De Vylder and Goovaerts [10] proved that a similar formula holds too for a compound Poisson model with continuous claim amounts; see also Ignatov and Kaishev [18].…”