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

2007 ◽  
Vol 39 (2) ◽  
pp. 385-406 ◽  
Author(s):  
Susan M Pitts ◽  
Konstadinos Politis
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Deficit At Ruin ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
Special Cases ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
Expected Discounted Penalty

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) defined the function mδ(u) =E[e−δ τ(u)w(X+(u), Y+(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that mδ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for mδ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired mδ(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.

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

2007 ◽  
Vol 39 (02) ◽  
pp. 385-406
Author(s):  
Susan M Pitts ◽  
Konstadinos Politis
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Deficit At Ruin ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
Special Cases ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
Expected Discounted Penalty

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) defined the function m δ(u) =E[e−δ τ(u) w(X +(u), Y +(u)) 1 (τ(u) &lt; ∞)], 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.

On the Expected Discounted Penalty Function for a Risk Model with Thinning Process

2010 ◽  
Vol 29-32 ◽  
pp. 1156-1161
Author(s):  
Wen Guang Yu
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Standard Wiener Process ◽  
Deficit At Ruin ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
Special Cases ◽  
Discounted Penalty Function ◽  
Thinning Process ◽  
Expected Discounted Penalty

This paper studies the expected discounted penalty function for a risk model in which the arrival of insurance policies is a Poisson process and the process of claim occurring is -thinning process. Using backward differential argument, we derive the integro-differential equation satisfied by the expected discounted penalty function when the stochastic discount interest process is perturbed by standard Wiener process and Poisson-Geometric process. Applications of the integral equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. In some special cases with exponential distributions, closed form expressions for these quantities are obtained.

scholarly journals Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

2007 ◽  
Vol 37 (02) ◽  
pp. 293-317 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
Steve Drekic
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Probability Distributions ◽  
Computational Procedure ◽  
Deficit At Ruin ◽  
Time Of Ruin ◽  
Special Cases ◽  
The Deficit At Ruin ◽  
Algorithmic Analysis ◽  
The Time Of Ruin

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

scholarly journals Some Explicit Solutions for the Joint Density of the Time of Ruin and the Deficit at Ruin

2008 ◽  
Vol 38 (1) ◽  
pp. 259-276 ◽  
Author(s):  
David C.M. Dickson
Keyword(s):  
Risk Model ◽  
Joint Density ◽  
Explicit Solutions ◽  
Classical Risk Model ◽  
Exponential Distributions ◽  
Deficit At Ruin ◽  
Time Of Ruin ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
The Classical Risk Model

Using probabilistic arguments we obtain an integral expression for the joint density of the time of ruin and the deficit at ruin. For the classical risk model, we obtain the bivariate Laplace transform of this joint density and invert it in the cases of individual claims distributed as Erlang(2) and as a mixture of two exponential distributions. As a consequence, we obtain explicit solutions for the density of the time of ruin.

scholarly journals Some Explicit Solutions for the Joint Density of the Time of Ruin and the Deficit at Ruin

2008 ◽  
Vol 38 (01) ◽  
pp. 259-276 ◽  
Author(s):  
David C.M. Dickson
Keyword(s):  
Risk Model ◽  
Joint Density ◽  
Explicit Solutions ◽  
Classical Risk Model ◽  
Exponential Distributions ◽  
Deficit At Ruin ◽  
Time Of Ruin ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
The Classical Risk Model

Using probabilistic arguments we obtain an integral expression for the joint density of the time of ruin and the deficit at ruin. For the classical risk model, we obtain the bivariate Laplace transform of this joint density and invert it in the cases of individual claims distributed as Erlang(2) and as a mixture of two exponential distributions. As a consequence, we obtain explicit solutions for the density of the time of ruin.

scholarly journals Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

2007 ◽  
Vol 37 (2) ◽  
pp. 293-317 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
Steve Drekic
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Probability Distributions ◽  
Computational Procedure ◽  
Deficit At Ruin ◽  
Time Of Ruin ◽  
Special Cases ◽  
The Deficit At Ruin ◽  
Algorithmic Analysis ◽  
The Time Of Ruin

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

scholarly journals On the Expected Discounted Penalty Function for the Classical Risk Model with Potentially Delayed Claims and Random Incomes

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Huiming Zhu ◽  
Ya Huang ◽  
Xiangqun Yang ◽  
Jieming Zhou
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Claim Amount ◽  
Classical Risk Model ◽  
Main Claim ◽  
Expected Discounted Penalty Function ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
The Classical Risk Model ◽  
Expected Discounted Penalty

We focus on the expected discounted penalty function of a compound Poisson risk model with random incomes and potentially delayed claims. It is assumed that each main claim will produce a byclaim with a certain probability and the occurrence of the byclaim may be delayed depending on associated main claim amount. In addition, the premium number process is assumed as a Poisson process. We derive the integral equation satisfied by the expected discounted penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the expected discounted penalty function is derived. Finally, for the exponential claim sizes, we present the explicit formula for the expected discounted penalty function.

Improvement to the Expected Discounted Penalty Function for a Classical Risk Model with a Threshold Dividend Strategy

2010 ◽  
Vol 29-32 ◽  
pp. 1150-1155
Author(s):  
Wen Guang Yu ◽  
Zhi Liu
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Penalty Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Expected Discounted Penalty Function ◽  
Threshold Dividend Strategy ◽  
Interest Force ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

In this paper, we study the expected discounted penalty function for a classical risk model in which a threshold dividend strategy is used for a classical risk model and the discount interest force process is not a constant, but a stochastic process driven by Poisson process and Wiener process. In this model, we derive and solve an integro-differential equation for the expected discounted penalty function.

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