A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

Ruan, Jiechang; Yu, Wenguang; Song, Ke; Sun, Yuan; Huang, Yubing; Yu, Xinliang

doi:10.3390/math7100891

Cited by 7 publications

(9 citation statements)

References 39 publications

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“…Substitute (32) into (31) and then combine the yielding equation with (30), one can find that V ξ ′ (b− ) � V ξ ′ (b+) coincides with (26).…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

“…e concavity of V ξ in Figures 1(b) and 1(c) motivates us to find the optimal threshold (denote it by b * ). By (26), we can get the corresponding optimal threshold level b * for different cases, as shown in Table 1.…”

Section: Special Casesmentioning

confidence: 99%

“…Wang and Zhang [22] studied the optimal loss-carry-forward tax problem with the general draw-down stopping. For application of the draw-down time in risk theory, we are referred to Wang and Ming [23], Wang et al [24], Chen et al [25], Avram et al [17], Ruan et al [26], Yu et al [27], and Zhang et al [28].…”

Section: Introductionmentioning

confidence: 99%

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A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

Zhang

Liu

2020

Mathematical Problems in Engineering

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This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

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“…Substitute (32) into (31) and then combine the yielding equation with (30), one can find that V ξ ′ (b− ) � V ξ ′ (b+) coincides with (26).…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Section: Special Casesmentioning

confidence: 99%

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A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

Zhang

Liu

2020

Mathematical Problems in Engineering

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“…Since then, it has become a standard tool for studying ruin related quantities. We refer the interested readers to Lin et al [40], Huang and Yu [41], Ruan et al [42], Li et al [43], Wang et al [44], Yang et al [45], Yu [46,47], Yuen et al [48], Huang et al [49], Xie and Zou [50].…”

Section: Introductionmentioning

confidence: 99%

On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model

Guo

Wang

et al. 2020

Mathematics

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In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical risk model. We derive the equations and the boundary conditions satisfied by the Gerber-Shiu function, the expected discounted capital injection function and the expected discounted dividend function by assuming that the observation interval and claim amount are exponentially distributed, respectively. Numerical examples are also given to further analyze the influence of relevant parameters on the actuarial function of the risk model.

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“…Yang and Zhang [4], Wang [5], and Cao and Wan [6] studied the optimal reinsurance and investment problems of expected utility maximization. e latest researches on insurance and investment management problem can be referred to Yu et al [7], Zhang et al [8], Peng et al [9], Yu et al [10], Ruan et al [11], Yu et al [12], Huang et al [13], Zeng et al [14], Li et al [15] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Optimal Reinsurance-Investment Problem under a CEV Model: Stochastic Differential Game Formulation

Chen

2020

Mathematical Problems in Engineering

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This paper focuses on a stochastic differential game played between two insurance companies, a big one and a small one. In our model, the basic claim process is assumed to follow a Brownian motion with drift. Both of two insurance companies purchase the reinsurance, respectively. The big company has sufficient asset to invest in the risky asset which is described by the constant elasticity of variance (CEV) model and acquire new business like acting as a reinsurance company of other insurance companies, while the small company can invest in the risk-free asset and purchase reinsurance. The game studied here is zero-sum where there is a single exponential utility. The big company is trying to maximize the expected exponential utility of the terminal wealth to keep its advantage on surplus while simultaneously the small company is trying to minimize the same quantity to reduce its disadvantage. In this paper, we describe the Nash equilibrium of the game and prove a verification theorem for the exponential utility. By solving the corresponding Fleming-Bellman-Isaacs equations, we derive the optimal reinsurance and investment strategies. Furthermore, numerical examples are presented to show our results.

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A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

Cited by 7 publications

References 39 publications

A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model

Optimal Reinsurance-Investment Problem under a CEV Model: Stochastic Differential Game Formulation

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