On a Generalization from Ruin to Default in a Lévy Insurance Risk Model

Feng, Runhuan; Shimizu, Yasutaka

doi:10.1007/s11009-012-9282-y

Cited by 21 publications

(21 citation statements)

References 31 publications

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“…Feng and Shimizu (2013) extends the analysis of this new class of functions to the general spectrally negative Lévy risk process. In particular, Feng and Shimizu (2013, Proposition 5.1) tells that the Gerber-Shiu function f −δ (x; w 0 , w) in (14) is indeed a special case of H 0,δ (x) given in (15), for an appropriate choice of l depending on w 0 and w. From the statistical side, a nonparametric estimator of the Gerber-Shiu function for the risk process (1) is suggested by Shimizu (2012).…”

Section: The Double Laplace Transform Of the Time Of Ruin T And The Imentioning

confidence: 91%

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

Gatto

Baumgartner

2014

Methodol Comput Appl Probab

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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...

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Section: The Double Laplace Transform Of the Time Of Ruin T And The Imentioning

confidence: 91%

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

Gatto

Baumgartner

2014

Methodol Comput Appl Probab

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“…This DRE is essential to construct an estimator of φ as is seen below. The condition (1) is necessary to get the DRE, and we cannot include the case where 1 0 z ν(dz) = ∞ in this statement; see Feng and Shimizu (2013), Lemma 3.1 and its remark.…”

Section: The Laguerre Expansion Of φmentioning

confidence: 99%

“…(2) compatibility with financial theories and dynamical risk managements; (3) statistical prediction of the future surplus. On the aspect (1): Lévy process has properties of independent and stationary increments, and it derives many beautiful mathematical formulae for ruin probability and other ruin-related quantities via the fluctuation theory of Lévy processes; see Huzak et al (2004), Feng and Shimizu (2013), and Kyprianou (2014), among others. On (2), Trufin et al (2011), Shimizu and Tanaka (2018) proposed dynamic risk measures based on ruin probability and its related quantities, which are useful not only in insurance but also financial mathematics.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples

Shimizu

Zhang

2019

Risks

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A statistical inference for ruin probability from a certain discrete sample of the surplus is discussed under a spectrally negative Lévy insurance risk. We consider the Laguerre series expansion of ruin probability, and provide an estimator for any of its partial sums by computing the coefficients of the expansion. We show that the proposed estimator is asymptotically normal and consistent with the optimal rate of convergence and estimable asymptotic variance. This estimator enables not only a point estimation of ruin probability but also an approximated interval estimation and testing hypothesis.

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“…This operator-based approach appeared in various forms in the context of different risk models such as Cai et al (2009) andFeng (2009b). In a forthcoming paper by Feng and Shimizu (2010), the operator-based approach is further extended to analyze a more general Lévy risk model. It should be pointed out that a subset of the operator identities shown in Appendix B has been used in Albrecher et al (2010) as a basis for automated computer algebra systems, such as Mathematica, for solving boundary value problems in the context of a renewal risk model.…”

Section: Introductionmentioning

confidence: 99%

On a Generalization from Ruin to Default in a Lévy Insurance Risk Model

Cited by 21 publications

References 31 publications

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

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

Asymptotically Normal Estimators of the Ruin Probability for Lévy Insurance Surplus from Discrete Samples

An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models

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