On moments based Padé approximations of ruin probabilities

Abstract

cited By (since 1996)2International audienceIn this paper, we investigate the quality of the moments based Pad approximation of ultimate ruin probabilities by exponential mixtures. We present several numerical examples illustrating the quick convergence of the method in the case of Gamma processes. While this is not surprising in the completely monotone case (which holds when the shape parameter is less than 1), it is more so in the opposite case, for which we improve even further the performance by a fix-up which may be of special importance due to its potential use in the four moments Gamma approximation. We also review the connection of the exponential mixtures approximation to Pad approximation, orthogonal polynomials, and Gaussian quadrature. These connections may turn out useful for providing rates of convergence. © 2011 Elsevier B.V. All rights reserved

On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes

“…In the case of insufficient data, Padé approximations of the output function seem to deserve special attention, because they involve only a few input moments, and seem to extract at low orders the essence of the data. Note, as pointed out in Avram et al (2011Avram et al ( , 2018; Avram and Pistorius (2014), that the classical ruin theory approximations of Cramér-Lundberg, De Vylder and Renyi are all first order Padé approximations. Slightly increasing the order yields more sophisticated moments based approximations Avram et al (2018); Avram and Pistorius (2014); Ramsay (1992).…”

“…In the context of probability distributions, given a density function f (x) and its Laplace transform f (s), the inverse Laplace transform of the order (m, m) Padé approximant of f (s) provides a matrix exponential approximation of f (x) that matches the first 2m moments of f (x) (including m 0 ). In Avram et al (2011) this approach was used to approximate ruin probabilities. In this paper we develop the same approach to approximate the scale function W q (x) (Section 3).…”

“…The preceding observations make Laplace inversion by rational approximation of the Laplace transform a quite attractive tool in ruin theory and related fields, from both a pedagogical and numerical point of view Avram et al (2011Avram et al ( , 2018; Avram and Pistorius (2014). One drawback is the appearance of non-admissible roots (for example, with (γ i ) < 0), which will imply non-admissible ruin probabilities for very large initial reserves x.…”

Regularization and error estimate of infinite‐time ruin probabilities for Cramer‐Lundberg model

“…This model embraces a huge literature in mathematical area. Beekman and Bowers, De Vylder, Ramsay, Garcia, Avram et al, Tran, etc proposed many different methods to compute ultimate and finite‐time ruin probabilities of insurance companies.…”

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].…”