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

He, Yue; Kawai, Reiichiro; Shimizu, Yasutaka; Yamazaki, Kazutoshi

doi:10.1016/j.insmatheco.2022.12.003

Cited by 9 publications

(4 citation statements)

References 226 publications

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“…Now the function may be instrumental in understanding the vulnerability of an insurance institution and has been generalized in the literature in various models, see [12][13][14][15][16][17][18][19][20] for more details. He et al [21] also provided a comprehensive review of existing works for the Gerber-Shiu function from practical perspectives.…”

Section: Introductionmentioning

confidence: 99%

On a Perturbed Risk Model with Time-Dependent Claim Sizes

Wei,

Hao,

Song

et al. 2024

Journal of Mathematics

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We consider a risk model perturbed by a Brownian motion, where the individual claim sizes are dependent on the inter-claim times. We study the Gerber–Shiu functions when ruin is due to a claim or the jump-diffusion process. Integro-differential equations and Laplace transforms satisfied by the Gerber–Shiu functions are obtained. Then, it is shown that the expected discounted penalty functions satisfy defective renewal equations. Explicit expressions can be obtained for exponential claim sizes. Finally, a numerical example is provided to measure the impact of the various dependence parameters in the risk model on the ruin probabilities.

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Section: Introductionmentioning

confidence: 99%

On a Perturbed Risk Model with Time-Dependent Claim Sizes

Wei,

Hao,

Song

et al. 2024

Journal of Mathematics

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“…[7] introduced the expected discounted penalty function (referred to as the Gerber-Shiu function), which is used to analyze other functions related to the ruin event. Since then, the Gerber-Shiu function has become a unified tool for determining the quantities associated with ruin, and a large number of ruin theory studies have been devoted to the Gerber-Shiu function, which has been widely studied and extended to different surplus processes (see, e.g., [8][9][10][11]). The renewal risk model allows for a more general distribution of the interclaim time, and several researchers have contributed to the analysis of the Sparre Andersen risk model by studying the Gerber-Shiu function (see, e.g., [9,10]).…”

Section: Introductionmentioning

confidence: 99%

Joint Moments of Surplus Immediately Before Ruin and Deficit at Ruin in the Renewal Risk Model

Wang,

Wang

2024

JAMC

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This paper studies the risk process where the individual claim amount density and the interclaim time density are assumed to be arbitrary. It uses the renewability of the risk process at the time of claims and the cumulative law of mathematical expectation to introduce a defective renewal equation satisfied by the joint moments of the surplus immediately before ruin and the deficit at ruin. From this equation, an explicit expression for the above joint moments (including the ruin probability) is obtained by the Laplace transform. Explicit expressions for the ruin probability are given for the case where the claim amount distribution is exponential, and numerical examples are given to analyze the effect of the relevant parameters on the ruin probability under the conditions that the interclaim times follow a Gamma distribution and a generalized Gamma distribution, respectively.

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“…Te expected discounted penalty function provides a unifed framework for the evaluation of various risk quantities. For a systematic study of the Gerber-Shiu theory, one can refer to references [1][2][3][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…Although the expected discounted penalty function was proposed more than two decades ago and continues to play an important role in actuarial science research, there are still many unsolved problems such as the two here for generalizing the discount rate in this function from a constant to a random variable for the nonclassical risk model and looking for an efective numerical scheme for this function with no explicit solution (see [3,[5][6][7][8][9][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%

On the Expected Discounted Penalty Function Using Physics-Informed Neural Network

Wang,

Wang

2023

Journal of Mathematics

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We study the expected discounted penalty at ruin under a stochastic discount rate for the compound Poisson risk model with a threshold dividend strategy. The discount rate is modeled by a Poisson process and a standard Brownian motion. By applying the differentiation method and total expectation formula, we obtain an integrodifferential equation for the expected discounted penalty function. From this integrodifferential equation, a renewal equation and an asymptotic formula satisfied by the expected discounted penalty function are derived. In order to solve the integrodifferential equation, we use a physics-informed neural network (PINN) for the first time in risk theory and obtain the numerical solutions of the expected discounted penalty function in some special cases of the penalty at ruin.

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

Cited by 9 publications

References 226 publications

On a Perturbed Risk Model with Time-Dependent Claim Sizes

On a Perturbed Risk Model with Time-Dependent Claim Sizes

Joint Moments of Surplus Immediately Before Ruin and Deficit at Ruin in the Renewal Risk Model

On the Expected Discounted Penalty Function Using Physics-Informed Neural Network

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