Distributional study of finite-time ruin related problems for the classical risk model

Li, Shuanming; Lu, Yi

doi:10.1016/j.amc.2017.07.054

Cited by 6 publications

(4 citation statements)

References 23 publications

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“…This function has become known as the Gerber‐Shiu function, and it has created a unified approach for determining the time to ruin, probability of ruin, deficit at ruin, surplus before ruin, loss‐causing claim amount, and other quantities of interest under ruin conditions. In this connection, Li and Dickson, Cheung and Landriault, Cheung et al, Biffis and Morales, and Li and Lu presented some interesting results concerning the maximum surplus, minimum surplus, and maximum deficit problems under ruin conditions.…”

Section: Introductionmentioning

confidence: 93%

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable, dependent claim occurrences

Gebizlioğlu

Eryılmaz

2018

Appl Stoch Models Bus & Ind

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This paper investigates a discrete-time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite-time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management-oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete-time compound binomial risk model, for which claim occurrences are independent and identically distributed. KEYWORDSbeta-binomial distribution, compound binomial model, dependence, economic capital, exchangeable random variables, maximum surplus, risk reserve 858

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Section: Introductionmentioning

confidence: 93%

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable, dependent claim occurrences

Gebizlioğlu

Eryılmaz

2018

Appl Stoch Models Bus & Ind

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“…(2.14) for a finite time period T > 0 [17,51], where w(•) models the penalty in the case of ruin whereas P(•) models the reward for survival. This is further generalized with two variables [94,106,107] as in…”

Section: Finite Time Horizonmentioning

confidence: 99%

“…After its first development [65] for the Cramér-Lundberg model (2.18), the Laplace transform based method has been employed, extended and improved in a wide variety of problem settings. Some examples include dependence between interclaim arrivals and claim sizes [24,124] together with perturbation [196], perturbation with two-sided jumps [198], perturbation and investment with penalty and reward in a finite time [17], delayed claims induced by main claims [177,207], random income modeled by a compound Poisson process [6] and additionally with delayed-claims [58,204], constant interests that can be positive or negative [139], capital injection restoring the surplus to a certain level [50], a finite-time problem with and without perturbation [106,107], and a generalized stochastic income with additional discounting [167]. Those studies succeed to obtain the the Laplace transform of the Gerber-Shiu function, or even the Gerber-Shiu function itself in explicit form by the inverse Laplace transform.…”

Section: Cramér-lundberg and Lévy Modelsmentioning

confidence: 99%

The Gerber-Shiu discounted penalty function: A review from practical perspectives

He¹,

Kawai²,

Shimizu³

et al. 2022

Preprint

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The Gerber-Shiu function provides a unified framework for the evaluation of a variety of risk quantities. Ever since its establishment, it has attracted constantly increasing interests in actuarial science, whereas the conventional research has been focused on finding analytical or semi-analytical solutions, either of which is rarely available, except for limited classes of penalty functions on rather simple risk models. In contrast to its great generality, the Gerber-Shiu function does not seem sufficiently prevalent in practice, largely due to a variety of difficulties in numerical approximation and statistical inference. To enhance research activities on such implementation aspects, we provide a full review of existing formulations and underlying surplus processes, as well as an extensive survey of analytical, semi-analytical and asymptotic methods for the Gerber-Shiu function, which altogether shed fresh light on its numerical methods and statistical inference for further developments. On the basis of an exhaustive collection of 207 references, the present survey can serve as an insightful guidebook to model and method selection from practical perspectives as well.

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“…In insurance risk theory, the study of finite-time ruin problems dates back to Prahbu [24] and Seal [26], where the finite-time ruin probability (or survival probability) is studied under the compound Poisson model. Over the last few decades, a series of contributions have been made by actuarial researchers; see, for example, Dickson and Willmot [13], Dickson and Li [12], Li and Sendova [20], Kuznetsov and Morales [17], Li and Lu [19] and Li et al [21]. In particular, the finite-time ruin probability and the finite-time Gerber-Shiu function are studied in Lee et al [18] and Li et al [22] by the Fourier-cosine series expansion method, respectively.…”

Section: Introductionmentioning

confidence: 99%

Gerber-Shiu analysis in the compound Poisson model with constant inter-observation times

Xie

Yu²,

Zhang³

et al. 2022

Prob. Eng. Inf. Sci.

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In this paper, the classical compound Poisson model under periodic observation is studied. Different from the random observation assumption widely used in the literature, we suppose that the inter-observation time is a constant. In this model, both the finite-time and infinite-time Gerber-Shiu functions are studied via the Laguerre series expansion method. We show that the expansion coefficients can be recursively determined and also analyze the approximation errors in detail. Numerical results for several claim size density functions are given to demonstrate effectiveness of our method, and the effect of some parameters is also studied.

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Distributional study of finite-time ruin related problems for the classical risk model

Cited by 6 publications

References 23 publications

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable, dependent claim occurrences

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable, dependent claim occurrences

The Gerber-Shiu discounted penalty function: A review from practical perspectives

Gerber-Shiu analysis in the compound Poisson model with constant inter-observation times

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