On the discounted penalty function in the renewal risk model with general interclaim times

Willmot, Gordon E.

doi:10.1016/j.insmatheco.2006.08.005

Cited by 80 publications

(61 citation statements)

References 13 publications

(11 reference statements)

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“…This function is known to satisfy a defective renewal equation (Gerber and Shiu 1998;Lin and Garrido 2004;Willmot 2007) but easy explicit formulae for φ (u) are only available for certain special cases for the claim size distribution Willmot 1999, 2000;Landriault and Willmot 2008). Let w(x, y) = 1; we then obtain the expression for the defective Laplace transform (LT) of the time of ruin φ (u) = E e −δ T I (T < ∞) |R (0) = u , and if in addition δ = 0, then φ (u) = P [T < ∞|R (0) = u] = ψ (u), i.e.…”

Section: Assumptions and Preliminariesmentioning

confidence: 99%

Ruin probability and time of ruin with a proportional reinsurance threshold strategy

Castañer

Bielsa

Marmol

2010

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In this paper we present a threshold proportional reinsurance strategy and we analyze the effect on some solvency measures: ruin probability and time of ruin. This dynamic reinsurance strategy assumes a retention level that is not constant and depends on the level of the surplus. In a model with inter-occurrence times generalized Erlang(n)-distributed we obtain the integro-differential equation for the Gerber-Shiu function. Then, we present the solution for inter-occurrence times exponentially distributed and claim amount phase-type(N). Some examples for exponential and phase-type(2) claim amount are presented. Finally, we show some comparisons between threshold reinsurance and proportional reinsurance.

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Section: Assumptions and Preliminariesmentioning

confidence: 99%

Ruin probability and time of ruin with a proportional reinsurance threshold strategy

Castañer

Bielsa

Marmol

2010

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“…3 have been used independently by Willmot (2007), Vlasiou (2007) and Lefévre (2007). In fact, in the renewal model of risk theory, Willmot (2007), see also Landriault and Willmot (2008), used a similar factorization as Eq. 3 for the tail of claim size distribution and studied some analytical expressions for the Gerber-Shiu discounted penalty function.…”

Section: Preliminaries and Examplesmentioning

confidence: 99%

An Operator Property of the Distribution of a Nonhomogeneous Poisson Process with Applications

Psarrakos

2015

Methodol Comput Appl Probab

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“…3, then following the ideas as in [35,37,49], for δ 2 > 0, L δ 2 ,X (·) is seen to be a linear combination of exponential terms. Hence, all quantities on the right-hand side of (4.6) can be explicitly evaluated, and the choice of penalty function w * (·) is also a linear combination of exponential terms.…”

Section: Remarkmentioning

confidence: 99%

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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On the discounted penalty function in the renewal risk model with general interclaim times

Cited by 80 publications

References 13 publications

Ruin probability and time of ruin with a proportional reinsurance threshold strategy

Ruin probability and time of ruin with a proportional reinsurance threshold strategy

An Operator Property of the Distribution of a Nonhomogeneous Poisson Process with Applications

On a class of stochastic models with two-sided jumps

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