Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

Abstract

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by BarndorffNielsen and Schmidli (1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute. Key wordsConditional distribution, cumulant generating function, Gerber-Shiu function, importance sampling, Laplace transform, large deviations techniques, Monte Carlo simulation, relative error. IntroductionThis article considers the risk process perturbed by diffusion as a stochastic model for the fluctuations of the insurer reserve over the time. Let X 1 , X 2 , . . . > 0 be independent individual claim amounts with absolutely continuous distribution function F X , let {N t } t ≥0 be a Poisson process with intensity λ > 0 and let {W t } t ≥0 be a Wiener process. The individual claim amounts, the Poisson process and the Wiener process are assumed independent. All these random elements are defined on the probability space (Ω, F , P).The Cramér-Lundberg perturbed risk process is then defined by (1991). There is a considerable literature on perturbed risk processes and a review is provided by Schmidli (1999). Two extensive references on risk processes are Asmussen and Albrecher (2010) and Rolski et al. (1999). Note that (1) is an instance of a spectrally negative Lévy process.Let us define the time of ruin asThe probability of ruin within the finite time horizon (0, t ), for any tIt is the probability that {Y t } t ≥0 falls below the zero line prior to time t . The probability of ruin within the infinite time horizon is defined by ψ(x) = P(T < ∞ | Y 0 = x). It is the probability that {Y t } t ≥0ever falls below the zero line. In the following, unless the time horizon is stated explicitly, the probability of ruin refers to the infinite time horizon. The quantity of interest in this article is the probability of ruin within a finite time horizon and the aim is to provide an accurate and efficient computational technique for this probability, by generalizing the saddlepoint approximation of Barndorff-Nielsen and Schmidli (1995) to the perturbed risk process (1). This approximation is then compared with importance sampling based on exponential change of measure. For this purpose, a generalization of the importance sampling algorithm for the unperturbed risk process, given by Asmussen (1985) and Asmussen and Albrecher (2010, Section XV.4), is provided. Although computer intensive, importance sampling is typically very accurate, see e. g. Gatto and Mosimann (2012) for the infini...

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

“…Most relative deviations between importance sampling and saddlepoint approximation are below 5% and this coincidence between these two different techniques gives good evidence of high accuracy for these two alternative methods. More comparisons can be found in Gatto and Baumgartner (2014). A numerical comparison for the infinite time horizon probability of ruin can be found in Gatto and Mosimann (2012), where it is again shown that importance sampling and saddlepoint approximation have very small relative deviations.…”

“…We compute directly the relative deviations of importance sampling w.r.t. the saddlepoint approximation of Gatto and Baumgartner (2014), which is another large deviations technique. Similar approximations are thus expected with both methods.…”

“…The saddlepoint approximation in this context originates from Daniels (1954) and Lugannani and Rice (1980). Gatto and Mosimann (2012) and Gatto and Baumgartner (2014) provide comparisons between importance sampling and the saddlepoint approximation, for probabilities of ruin in finite and infinite time horizons, however only for the compound Poisson process with Wiener perturbation. In this article we consider the more general spectrally negative Lévy risk processes.…”

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

“…For the particular situation where the loss process S is a compound Poisson process perturbed by a Wiener process, the desired quantityψ(x, t) can be alternatively computed by the saddlepoint approximation to ψ(x, t) suggested by Gatto and Baumgartner (2014), together with the saddlepoint approximation to ζ(x, t) of Gatto (2010). Saddlepoint approximations are substantially faster to compute than importance sampling, although they are conceptually more sophisticated and by far less popular than Monte Carlo methods.…”

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.…”