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

Feng, Runhuan

doi:10.1016/j.insmatheco.2010.12.003

Cited by 17 publications

(14 citation statements)

References 23 publications

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“…Added to the list as shown in this paper include the moments of discounted claim costs up to ruin and the moments of operating costs up to ruin in a risk model with Markovian claim arrivals as described in Section 1.2. The function H was also studied by Feng (2009a,b) for a renewal risk model with phase-type interclaim times and used in Feng (2011) and Feng and Shimizu (2013a,b) to obtain potential measures of jump diffusion processes and Lévy processes.…”

Section: Generalizations Of the Gerber-shiu Functionmentioning

confidence: 99%

A unified analysis of claim costs up to ruin in a Markovian arrival risk model

Cheung

Feng

2013

Insurance: Mathematics and Economics

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An insurance risk model where claims follow a Markovian arrival process (MArP) is considered in this paper. It is shown that the expected present value of total operating costs up to default H, as a generalization of the classical Gerber-Shiu function, contains more non-trivial quantities than those covered in Cai et al. (2009), such as all moments of the discounted claim costs until ruin. However, it does not appear that the Gerber-Shiu function φ with a generalized penalty function which additionally depends on the surplus level immediately after the second last claim before ruin (Cheung et al. (2010a)) is contained in H. This motivates us to investigate an even more general function Z from which both H and φ can be retrieved as special cases. Using a matrix version of Dickson-Hipp operator (Feng (2009b)), it is shown that Z satisfies a Markov renewal equation and hence admits a general solution. Applications to other related problems such as the matrix scale function, the minimum and maximum surplus levels before ruin are given as well.

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Section: Generalizations Of the Gerber-shiu Functionmentioning

confidence: 99%

A unified analysis of claim costs up to ruin in a Markovian arrival risk model

Cheung

Feng

2013

Insurance: Mathematics and Economics

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“…Taking the derivative with respect to u yields the IDE Equation (12). The proof of the boundary condition in Equation (13) mostly follows that in [24] (Theorem 2.1), although it is a bit more tedious.…”

Section: Ide and Boundary Condition Formentioning

confidence: 95%

“…Recently, there has been increased interest in the aggregate claims until the ruin time of the underlying risk process (instead of a fixed time). Specifically, under the (perturbed) compound Poisson model and the phase-type renewal models, [8,9] studied the distribution of the aggregate claims until ruin without discounting; whereas [10] (Section 6), [11] (Section 4.2) and [12] (Section 5.2) analyzed the expected aggregate discounted claims until ruin. The higher moments of the aggregate discounted claims until ruin were also considered by [13] (Section 2.1) in a risk process with Markovian claims arrival (e.g., [14], Chapter XI.1), by [15] in a renewal risk model with arbitrary inter-claim times and by [16] in a dependent Sparre-Andersen risk model (e.g., [17]).…”

Section: Introductionmentioning

confidence: 99%

On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy

Cheung

Woo

2015

Risks

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In the compound Poisson insurance risk model under a dividend barrier strategy, this paper aims to analyze jointly the aggregate discounted claim amounts until ruin and the total discounted dividends until ruin, which represent the insurer's payments to its policyholders and shareholders, respectively. To this end, we introduce a Gerber-Shiu-type function, which further incorporates the higher moments of these two quantities. This not only unifies the individual study of various ruin-related quantities, but also allows for new measures concerning covariances to be calculated. The integro-differential equation satisfied by the generalized Gerber-Shiu function and the boundary condition are derived. In particular, when the claim severity is distributed as a combination of exponentials, explicit expressions for this Gerber-Shiu function in some special cases are given. Numerical examples involving the covariances between any two of (i) the aggregate discounted claims until ruin, (ii) the discounted dividend payments until ruin and (iii) the time of ruin are presented along with some interpretations.

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“…the Brownian motion risk model and compound Poisson model as special cases. While Zhang and Cheung [45] obtained the ruin-related quantities by solving integro-differential equations with the use of matrix Dickson-Hipp operators (see [15,18]), this paper expresses the results only in terms of scale functions and potential measures.…”

Section: Remarkmentioning

confidence: 99%

A note on a Lévy insurance risk model under periodic dividend decisions

Zhang¹,

Cheung²

2018

Journal of Industrial &Amp; Management Optimization

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In this paper, we consider a spectrally negative Lévy insurance risk process with a barrier-type dividend strategy. In contrast to the traditional barrier strategy in which dividends are payable to the shareholders immediately when the surplus process reaches a fixed level b (as long as ruin has not yet occurred), it is assumed that the insurer only makes dividend decisions at some discrete time points in the spirit of [1]. Under such a dividend strategy with Erlang inter-dividend-decision times, expressions for the Gerber-Shiu expected discounted penalty function proposed in [24] and the moments of total discounted dividends payable until ruin are derived. The results are expressed in terms of the scale functions of a spectrally negative Lévy process and an embedded spectrally negative Markov additive process. Our analyses rely on the introduction of a potential measure associated with an Erlang random variable. Numerical illustrations are also given.2010 Mathematics Subject Classification. Primary: 60G51, 60J75; Secondary: 91B30, 62P05.

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

Cited by 17 publications

References 23 publications

A unified analysis of claim costs up to ruin in a Markovian arrival risk model

A unified analysis of claim costs up to ruin in a Markovian arrival risk model

On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy

A note on a Lévy insurance risk model under periodic dividend decisions

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