Calculating Ruin Probabilities via Product Integration

Abstract

When claims in the compound Poisson risk model are from a heavy-tailed distribution (such as the Pareto or the lognormal), traditional techniques used to compute the probability of ultimate ruin converge slowly to desired probabilities. Thus, faster and more accurate methods are needed. Product integration can be used in such situations to yield fast and accurate estimates of ruin probabilities because it uses quadrature weights that are suited to the underlying distribution. Tables of ruin probabilities for the Pareto and lognormal distributions are provided.

A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments

“…Surprisingly few of the suggested methods take advantage of the vast knowledge about integral equations in the numerical literature, and as a consequence many do not pay much attention to the error rate inherent in any numerical method. An exception is the paper by Ramsay and Usabel (1997) where the method of product integration is used to solve the Volterra integral equation for the ruin probability.…”

Stochastic successive approximation method for assessing the insolvency risk of an insurance company

“…The ideas of [25] are developed in [26][27][28][29][30][31][32][33]. In contrast to [25][26][27][28][29][30][31][32][33], the method of successive approximations [19] is applied in this article directly to Eq. (2) for the nonruin probability of the initial risk process, its uniform convergence is proved, and uniform convergence rate estimates are obtained for any initial approximation.…”

Calculating multivariate ruin probabilities via Gaver–Stehfest inversion technique

“…As shown in Table 3, approximations then showed 15-19 precision digits. Table 2 Pareto claim sizeB( Table 4 Log-normal claims size Results can be compared with those in Dickson and Waters (1992), Ramsay (1992a,b) and Ramsay and Usábel (1997) where appropriate.…”

“…They were based on a discretization of some aspect of the risk process and derived recursive expressions; see for example Dickson (1989), Dickson et al (1995), Goovaerts and De Vylder (1984), Panjer (1986), Panjer and Wang (1993), Ramsay (1992b) and Ramsay and Usábel (1997). Panjer and Wang (1993) describe the conditions under which these recursions are stable.…”

“…Though some of these recursive approaches may be able to determine x,y (u) to any desired degree of accuracy, they are not suitable for heavy-tailed distributions, such as the Pareto or lognormal distribution for two reasons, citing Ramsay and Usábel (1997):…”