Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

Abstract

This article provides importance sampling algorithms for computing the probabilities of various types ruin of spectrally negative Lévy risk processes, which are ruin over the infinite time horizon, ruin within a finite time horizon and ruin past a finite time horizon. For the special case of the compound Poisson process perturbed by diffusion, algorithms for computing probabilities of ruins by creeping (i.e. induced by the diffusion term) and by jumping (i.e. by a claim amount) are provided. It is shown that these algorithms have either bounded relative error or logarithmic efficiency, as t, x → ∞, where t > 0 is the time horizon and x > 0 is the starting point of the risk process, with y = t/x held constant and assumed either below or above a certain constant. Key words and phrasesBounded relative error; exponential tilt; Legendre-Fenchel transform; logarithmic efficiency; Lundberg conjugated measure; ruin due to creeping and to jump; ruin past a finite time horizon, within a finite and the infinite time horizons; saddlepoint approximation; short and long time horizons.

Non-Parametric Threshold Estimation for the Wiener–Poisson Risk Model

“…By renewal theory, they obtained the Pollaczeck-Khinchin formula of Φ(x). Accurate calculation and approximation for Ψ(x) has always been an inspiration and an important source of technological development for actuarial mathematics (see, e.g., [3][4][5][6][7][8][9]). Although various approximations to the probability of ruin (e.g., importance sampling or saddle-point approximations) are now available, developing alternative approximations of different nature is still an interesting and practical problem.…”

A logarithmic efficient estimator of the probability of ruin with recuperation for spectrally negative Lévy risk processes

“…Gatto (2014) provides importance sampling algorithms for finite and infinite time probabilities of ruin as well as for the probability of the ruin past a finite time horizon, in the context of spectrally negative Lévy processes. This article provides an importance sampling algorithm for the probability of ruin with recuperation for spectrally negative Lévy processes with light-tailed downwards jumps.…”