Gerber–Shiu analysis of a risk model with capital injections

Dickson, David; Qazvini, Marjan

doi:10.1007/s13385-016-0131-1

Cited by 13 publications

(12 citation statements)

References 22 publications

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“…Nie et al (2011) derived the ruin probability and the expected discounted capital injections until ruin, and applied these results to determine an optimal reinsurance contract that minimizes the ruin probability numerically. The finite-time ruin probability was then studied by Nie et al (2015), and this was extended by Dickson and Qazvini (2016) who further incorporated the number of claims until ruin into the analysis. Due to the spatial homogeneity of the compound Poisson process with constant premium rate, by shifting the process downward by b units, Nie et al (2011)'s model is also equivalent to one that restores the surplus level to zero if it falls between zero and −b but declares ruin if the surplus becomes less than −b.…”

Section: Introductionmentioning

confidence: 99%

On the Compound Poisson Risk Model With Periodic Capital Injections

Zhang¹,

Cheung

Yang³

2017

ASTIN Bull.

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The analysis of capital injection strategy in the literature of insurance risk models (e.g. Pafumi, 1998; Dickson and Waters, 2004) typically assumes that whenever the surplus becomes negative, the amount of shortfall is injected so that the company can continue its business forever. Recently, Nie et al. (2011) has proposed an alternative model in which capital is immediately injected to restore the surplus level to a positive level b when the surplus falls between zero and b, and the insurer is still subject to a positive ruin probability. Inspired by the idea of randomized observations in Albrecher et al. (2011b), in this paper, we further generalize Nie et al. (2011)'s model by assuming that capital injections are only allowed at a sequence of time points with inter-capital-injection times being Erlang distributed (so that deterministic time intervals can be approximated using the Erlangization technique in Asmussen et al. (2002)). When the claim amount is distributed as a combination of exponentials, explicit formulas for the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) and the expected total discounted cost of capital injections before ruin are obtained. The derivations rely on a resolvent density associated with an Erlang random variable, which is shown to admit an explicit expression that is of independent interest as well. We shall provide numerical examples, including an application in pricing a perpetual reinsurance contract that makes the capital injections and demonstration of how to minimize the ruin probability via reinsurance. Minimization of the expected discounted capital injections plus a penalty applied at ruin with respect to the frequency of injections and the critical level b will also be illustrated numerically.

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Section: Introductionmentioning

confidence: 99%

On the Compound Poisson Risk Model With Periodic Capital Injections

Zhang¹,

Cheung

Yang³

2017

ASTIN Bull.

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“…The main part of the known results on the Gerber-Shiu function is related with the Sparre Andersen model and various generalizations of this model. For instance, several cases of the Sparre Andersen model were considered by Dickson and Qazvini (2016), Landriault and Willmot (2008), Li and Garrido (2004), Li and Sendova (2015), Lin et al (2003), Schmidli (1999), Willmot and Dickson (2003). Properties of the Gerber-Shiu function in the risk renewal models perturbed by diffusion were investigated by Chi et al (2010), Tsai (2003), Tsai and Willmot (2002), Xu et al (2014), Zhang and Cheung (2016), Zhang et al (2012Zhang et al ( , 2017bZhang et al ( , 2014.…”

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confidence: 99%

The Gerber–Shiu Discounted Penalty Function for the Bi-Seasonal Discrete Time Risk Model

Navickienė¹,

Sprindys²,

Šiaulys³

2018

Informatica

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In this work, the discrete time risk model with two seasons is considered. In such model, the claims repeat with time periods of two units, i.e. claim distributions coincide at all even instants and at all odd instants. Our purpose is to derive an algorithm for calculating the values of the particular case of the Gerber-Shiu discounted penalty function E(e −δT 1 {T <∞}), where T is the time of ruin, and δ is a constant nonnegative force of interest. Theoretical results are illustrated by some numerical examples.

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“…We now generalise this result to the case u > 0, and the key reason for taking the following approach is that our proof also shows why P 1 k = 1 p k ðuÞ = ψðuÞ. From arguments in Dickson & Qazvini (2016) concerning the classical risk model with exponential claims, we can easily show that the probability generating function of N u for this model is…”

Section: Erlang (N) Risk Modelmentioning

confidence: 79%

“…Equation (1.1) has been used in relation to problems involving the number of claims until ruin in the classical risk model and variations thereof (see Dickson, 2012 andDickson &Qazvini, 2016). Generalised binomial series with integer values of τ > 2 appear in other ruin problems, and in this note we show how such series can be evaluated.…”

Section: Introductionmentioning

confidence: 99%

An identity based on the generalised negative binomial distribution with applications in ruin theory

Dickson

2018

Ann. actuar. sci.

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In this study, we show how expressions for the probability of ultimate ruin can be obtained from the probability function of the time of ruin in a particular compound binomial risk model, and from the density of the time of ruin in a particular Sparre Andersen risk model. In each case evaluation of generalised binomial series is required, and the argument of each series has a common form. We evaluate these series by creating an identity based on the generalised negative binomial distribution. We also show how the same ideas apply to the probability function of the number of claims in a particular Sparre Andersen model.

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Gerber–Shiu analysis of a risk model with capital injections

Cited by 13 publications

References 22 publications

On the Compound Poisson Risk Model With Periodic Capital Injections

On the Compound Poisson Risk Model With Periodic Capital Injections

The Gerber–Shiu Discounted Penalty Function for the Bi-Seasonal Discrete Time Risk Model

An identity based on the generalised negative binomial distribution with applications in ruin theory

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