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...
In this article we propose a bootstrap test for the probability of ruin in the compound Poisson risk process. We adopt the P-value approach, which leads to a more complete assessment of the underlying risk than the probability of ruin alone. We provide second-order accurate P-values for this testing problem and consider both parametric and nonparametric estimators of the individual claim amount distribution. Simulation studies show that the suggested bootstrap P-values are very accurate and outperform their analogues based on the asymptotic normal approximation. KEYWORDSEdgeworth expansion, exponential and log-normal claim amounts, normal approximation, P-value, pivotal quantity, resampling, second-order accuracy.
We analyze the insurer risk under the compound Poisson risk process perturbed by a Wiener process with infinite time horizon. In the first part of this article, we consider the capital required to have fixed probability of ruin as a measure of risk and then a coherent extension of it, analogous to the tail value at risk. We show how both measures of risk can be efficiently computed by the saddlepoint approximation. We also show how to compute the stabilities of these measures of risk with respect to variations of probability of ruin. In the second part of this article, we are interested in the computation of the probability of ruin due to claim and the probability of ruin due to oscillation. We suggest a computational method based on upper and lower bounds of the probability of ruin and we compare it to the saddlepoint and to the Fast Fourier transform methods. This alternative method can be used to evaluate the proposed measures of risk and their stabilities with heavy-tailed individual losses, where the saddlepoint approximation cannot be used. The numerical accuracy of all proposed methods is very high and therefore these measures of risk can be reliably used in actuarial risk analysis.Keywords Coherent measure of risk; Daniels' exponent; Fast Fourier transform; Lundberg's exponent; probability of ruin; probabilites of ruin due to claim and to oscillation; Richardson's extrapolation; saddlepoint approximation; stability; upper and lower bounds.
Focussing on the classical Cramér-Lundberg risk process, the main goal of this work is to provide an evaluation method for the insurer risk due to ruin over an infinite time horizon. In analogy to the well-known value of risk and tail value of risk, Gatto and Baumgartner [1] suggest the risk measures value at ruin and tail value at ruin for risk processes with additional Wiener perturbation as well as saddlepoint approximations to those risk measures. Since the special case with no perturbation is not discussed there, this is done here using the same ideas. In essence, the saddlepoint approximation method for obtaining quantiles provided by Wang [2] is combined with saddlepoint approximation results of .
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