On the Gerber–Shiu discounted penalty function in a risk model with two types of delayed-claims and random income

Gao, Jianwei; Wu, Lei

doi:10.1016/j.cam.2014.03.011

Cited by 11 publications

(6 citation statements)

References 15 publications

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“…In fact, the diffusion perturbation (Section 2.3.3) can also be interpreted in this way with Z = σW . Other examples in the literature include Poisson [19,20] and compound Poisson processes [29,58,92,201]. In the presence of upward jumps in the additional component, the resulting surplus process is no longer spectrally negative, while it can still be written as a Markov additive process (Section 2.3.5) as long as the positive jump size distribution is of phase-type [14].…”

Section: Surplus Processes With Additional Componentsmentioning

confidence: 99%

“…In the presence of upward jumps in the additional component, the resulting surplus process is no longer spectrally negative, while it can still be written as a Markov additive process (Section 2.3.5) as long as the positive jump size distribution is of phase-type [14]. The risk model containing by-claims induced by main claims can be modeled by setting this additional component to the negative of the running sum of by-claims [58,177,207].…”

Section: Surplus Processes With Additional Componentsmentioning

confidence: 99%

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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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Section: Surplus Processes With Additional Componentsmentioning

confidence: 99%

Section: Surplus Processes With Additional Componentsmentioning

confidence: 99%

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

He¹,

Kawai²,

Shimizu³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…They generalized the defective renewal equation for the expected discounted function of a penalty at the time of ruin in (Gerber and Landry 1998). (Gao and Wu 2014) worked on the Gerber-Shiu discounted penalty function in a risk model with two types of delayed-claims and random income. They developed a new delayed model with random premium income and two types of by-claims, and then derived an integral system of equations for the Gerber-Shiu discounted penalty function and explicit solution of the Laplace transform of the discounted penalty function.…”

Section: Introductionmentioning

confidence: 99%

Gerber–Shiu Function in a Class of Delayed and Perturbed Risk Model with Dependence

Adékambi

Takouda

2020

Risks

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This paper considers the risk model perturbed by a diffusion process with a time delay in the arrival of the first two claims and takes into account dependence between claim amounts and the claim inter-occurrence times. Assuming that the time arrival of the first claim follows a generalized mixed equilibrium distribution, we derive the integro-differential Equations of the Gerber–Shiu function and its defective renewal equations. For the situation where claim amounts follow exponential distribution, we provide an explicit expression of the Gerber–Shiu function. Numerical examples are provided to illustrate the ruin probability.

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“…Bao and Ye [6] and Labbé and Sendova [19] dealt with the Gerber-Shiu analysis in the compound Poisson risk model with compound Poisson premiums. Further exercises in risk processes with random incomes were done by Hao and Yang [17] and Jieming et al [18] in delayed claim strategies, Labbé et al [20] in a model with amount sizes to take positive as well as negative values, Gao and Wu [16] in a model with two classes of delayed claims and Shija and Jacob [26] in the Markov-modulated model.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent Phase-Type Claims and a Constant Barrier

Dibu

Jacob

2020

Infosys Science Foundation Series

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Inspired by the problems with random income feature, this paper focuses on an insurance risk model with MAP inter-arrival time for premiums as well as claims. We study the model for a convex combination of two types of interdependent Phase-type claims, where the probability of claim switching is directly associated with the inter-arrival time of claims. Furthermore, the surplus process of this model is assumed to be restricted by a horizontal barrier "b" above the initial surplus "u". The transient analysis of the corresponding Markovian fluid flow model is considered to develop the integral equations governing the Gerber-Shiu function and the expected discounted dividends paid until ruin. The closed-form solutions for these integral equations are obtained in terms of Lundberg roots. When the premium sizes are Phase-type distributed, the solutions are explicit at "u = b". For "u ≤ b", the solutions are explicit when the premium sizes are distributed exponentially. Finally, to validate and present the tractability of these solution expressions, some numerical illustrations are provided in individual cases.

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On the Gerber–Shiu discounted penalty function in a risk model with two types of delayed-claims and random income

Cited by 11 publications

References 15 publications

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

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

Gerber–Shiu Function in a Class of Delayed and Perturbed Risk Model with Dependence

Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent Phase-Type Claims and a Constant Barrier

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