The expected discounted penalty function under a risk model with stochastic income

Labbé, Chantal; Sendova, Kristina P.

doi:10.1016/j.amc.2009.07.049

Cited by 43 publications

(32 citation statements)

References 14 publications

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“…Note that we do not need to assume, as in Gerber and Shiu (2005), that the roots of the denominator are distinct. We end this section by treating a model considered by Labbé and Sendova (2008), see also the references in their paper. Suppose that the claim arrival process is a Poisson process.…”

Section: Lemma 2 (I) It Holds Thatmentioning

confidence: 99%

On the Gerber–Shiu function and change of measure

Schmidli¹

2010

Insurance: Mathematics and Economics

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Section: Lemma 2 (I) It Holds Thatmentioning

confidence: 99%

On the Gerber–Shiu function and change of measure

Schmidli¹

2010

Insurance: Mathematics and Economics

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“…For example, ruin probability results can be found in [10,41,48]. More generally, Labbé and Sendova [36] extended the study by showing that the Gerber-Shiu function satisfies a defective renewal equation and providing detailed analysis when the upward jumps follow an Erlang(n) distribution, whereas Albrecher et al [2] considered the Gerber-Shiu function (where the penalty depends on the deficit only) by making assumptions on either upward or downward jumps. We also refer interested readers to [7][8][9] for the study of risk models in which the two-sided jumps follow discrete distributions.…”

Section: Introductionmentioning

confidence: 98%

On a class of stochastic models with two-sided jumps

Cheung

2011

Queueing Syst

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In this paper a stochastic process involving two-sided jumps and a continuous downward drift is studied. In the context of ruin theory, the model can be interpreted as the surplus process of a business enterprise which is subject to constant expense rate over time along with random gains and losses. On the other hand, such a stochastic process can also be viewed as a queueing system with instantaneous work removals (or negative customers). The key quantity of our interest pertaining to the above model is (a variant of) the Gerber-Shiu expected discounted penalty function (Gerber and Shiu in N. Am. Actuar. J. 2(1):48-72, 1998) from ruin theory context. With the distributions of the jump sizes and their inter-arrival times left arbitrary, the general structure of the Gerber-Shiu function is studied via an underlying ladder height structure and the use of defective renewal equations. The components involved in the defective renewal equations are explicitly identified when the upward jumps follow a combination of exponentials. Applications of the Gerber-Shiu function are illustrated in finding (i) the Laplace transforms of the time of ruin, the time of recovery and the duration of first negative surplus in the ruin context; (ii) the joint Laplace transform of the busy period and the subsequent idle period in the queueing context; and (iii) the expected total discounted reward for a continuous payment stream payable during idle periods in a queue. Keywords

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“…And they derived a defective renewal equation satisfied by the Gerber-Shiu penalty function. Labbé and Sendova [5] considered a risk model where both premiums and claims follow compound Poisson processes. Soon, many papers on the risk model with random incomes were studied, see Yang and Zhang [11], Zhang and Yang [15], Yang and Hao [12], for instance.…”

Section: Introductionmentioning

confidence: 99%