Moments of the Surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process

Cheng, Yaonan; Tang, Qihe

doi:10.1080/10920277.2003.10596073

Cited by 63 publications

(59 citation statements)

References 18 publications

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“…where s is defined as in (5). Hence, in view of (A10), it can be seen that the integral in (A6), i.e., B k (x) also has a limit which admits the representation (A3) and obeys the system of equations (A4) which is established similarly, by passing to the limit in (A9), as…”

Section: (A5)mentioning

confidence: 84%

Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

Ignatov¹,

Kaishev

2011

Stochastics

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This is the accepted version of the paper.This version of the publication may differ from the final published version. A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of non-ruin of an insurance company, in a finite-time interval is derived, assuming independent, non-identically Erlang distributed claim inter-arrival times, Permanent repository link. ., any continuous joint distribution of the claim amounts and any non-negative, non-decreasing real function, representing its premium income. In the special case when τ i ∼ Erlang (g i , λ) , i = 1, 2, . . . it is shown that our main result yields a formula for the probability of non-ruin expressed in terms of the classical Appell polynomials. We give another special case of our non-ruin probability formula for. ., i.e., when the inter-arrival times are non-identically exponentially distributed and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when τ i ∼ Erlang(1, λ), i = 1, 2, . . .. The main result is extended further to a risk model in which inter-arrival times are dependent random variables, obtained by randomizing the Erlang shape or/and rate parameters. We give also some useful auxiliary results which characterize and express explicitly (and recurrently) the exponential Appell polynomials which appear in our finite time non-ruin probability formulae.

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Section: (A5)mentioning

confidence: 84%

Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

Ignatov¹,

Kaishev

2011

Stochastics

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“…We have considered the compound Poisson risk model. The literature has also included work on the expected discounted penalty function for the Sparre Andersen model, where the claims arrive in a renewal process (see, for example, Dickson and Hipp (2001), Cheng and Tang (2003), Gerber and Shiu (2005), and Li and Garrido (2004)). The functional here is more complicated than for the compound Poisson model, and is the subject of further research.…”

Section: Conclusion and Miscellaneous Remarksmentioning

confidence: 99%

Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

Pitts

Politis

2007

Adv. Appl. Probab.

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In the classical risk model with initial capital u, let τ (u) be the time of ruin, X + (u) be the risk reserve just before ruin, and Y + (u) be the deficit at ruin. Gerber and Shiu (1998) , where δ ≥ 0 can be interpreted as a force of interest and w(r, s) as a penalty function, meaning that m δ (u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for m δ (u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired m δ (u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

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“…Their results have triggered a widespread research on this topic. For recent results in this area, we refer to Dickson and Hipp [3] [4], Cheng and Tang [5], Gerber and Shiu [6] [7] [8], Li and Garrido [9] [10], Pitts and Polis [11] that have generalized the results for phase type distributions and for sub exponential distributions.…”

Section: Introductionmentioning

confidence: 99%

The Distribution of the Time of Ruin, the Surplus Immediately before Ruin and Deficit at Ruin under Two Sided Risk Renewal Process

Rebello¹,

Thampi²

2017

JMF

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In the renewal risk theory, the study of two sided jumps has been attracted by many researchers since its introduction. After the development of the distribution of modified inter time claim occurrence, the explicit expressions for ruin theory components in the literature under some assumptions, in this work, we examine probability density of the time of ruin, surplus immediately before ruin and deficit at ruin respectively under two sided risk process using some basic assumptions. Explicit expressions for distribution of interest are being derived.

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Moments of the Surplus before Ruin and the Deficit at Ruin in the Erlang(2) Risk Process

Cited by 63 publications

References 18 publications

Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

The Distribution of the Time of Ruin, the Surplus Immediately before Ruin and Deficit at Ruin under Two Sided Risk Renewal Process

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