On the efficient evaluation of ruin probabilities for completely monotone claim distributions

Abstract

In this paper we propose a highly accurate approximation procedure for ruin probabilities in the classical collective risk model, which is based on a quadrature/rational approximation procedure proposed by Trefethen et al. [12]. For a certain class of claim size distributions (which contains the completely monotone distributions) we give a theoretical justification for the method. We also show that under weaker assumptions on the claim size distribution, the method may still perform reasonably well in some cases. This in particular provides an efficient alternative to a related method proposed by Thorin [10]. A number of numerical illustrations for the performance of this procedure is provided for both completely monotone and other types of random variables. − 1 2π − e δu+ιxuL h (δ + ιx) dx. Note now thatL h (s) =L h (s), where w is the complex conjugate of w. Indeed, for Re(s) > 0 we have thatL h (s) = ∞ 0 e −st h(t) dt, from which the observation follows. For Re(s) ≤ 0 we may always find an s 0 such that Re(s 0 ) > 0 and |s − s 0 | < |s 0 |, from which

The Theory of Scale Functions for Spectrally Negative Lévy Processes

“…As we see from the table, the Talbot algorithm appears to be the most efficient. This is not surprising given the discussion in Section 5.3.3 where it was pointed out that this algorithm is well suited for processes with completely monotone jumps (see also results by Albrecher, Avram and Kortschak [5] Table 2: Computing W (q) (x) for the θ -process with σ = 0.…”

“…The spacing between the points in x-domain is chosen at δ x = 5/N x . In this way the grid x m = mδ x covers the whole interval [0,5] The results of this numerical experiment for computing W (q) (x) are presented in Table 1. We see that each of the Gaver-Stehfest/Euler/Talbot algorithms produce very accurate results.…”

“…These are the discretization of the Bromwich integral and Post-Widder formula coupled with Richardson extrapolation. Albrecher, Florin and Kortschak [5] develop algorithms to compute ruin probabilities for completely monotone claim distributions, which is equivalent to computing the scale function W (x) due to relation (2.8).…”

On moments based Padé approximations of ruin probabilities

“…Then, the ruin probabilities of the approximating process are computed using Thorin's integral formula (17) -see [26], which is valid when the shape parameter is smaller than 1 (completely monotone claims). However, we would like to note that the use of Thorin's formula is not essential, since rational approximation + partial fractions work also very well in this case (for example, the classic Padé approximation of Mathematica, or the one implemented here, as well as the rational approximations proposed in [27] and implemented in [28]). …”

Approximation of the ruin probability using the scaled Laplace transform inversion

“…See, for example, Gzyl et al [6], Avram et al [7], and Zhang et al [8] among others. A very interesting approximation based on the Trefethen–Weideman–Schmelzer (TWS) method (see [9]) is constructed in Albrecher et al [10]. In the latter work the authors assume that the claim size distribution represents a completely monotone function.…”