Interclaim Times Research Articles

Mathematical investigation of the Gerber–Shiu function in the case of dependent inter-claim time and claim size

In this paper we investigate the well-known Gerber–Shiu expected discounted penalty function in the case of dependence between the inter-claim times and the claim amounts. We set up an integral equation for it and we prove the existence and uniqueness of its solution in the set of bounded functions. We show that if δ > 0 , the limit property of the solution is not a regularity condition, but the characteristic of the solution even in the case when the net profit condition is not fulfilled. It is the consequence of the choice of the penalty function for a given density function. We present an example when the Gerber–Shiu function is not bounded, consequently, it does not tend to zero. Using an operator technique we also prove exponential boundedness.

On orderings and bounds in a generalized Sparre Andersen risk model

AbstractA generalization of the Gerber–Shiu function proposed by (Cheung et al., Scand. Actuarial J., in press, 2010) is used to derive some ordering properties for certain ruin‐related quantities in a Sparre Andersen type risk model. Additional bounds and/or refinements can be obtained by further assuming that the claim size and the interclaim time distributions possess certain reliability properties. Finally, numerical examples are considered to compare the exact solution to the bounds. Copyright © 2010 John Wiley & Sons, Ltd.

Analysis of ruin measures for the classical compound Poisson risk model with dependence

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.

Gerber–Shiu analysis with a generalized penalty function

A generalization of the usual penalty function is proposed, and a defective renewal equation is derived for the Gerber–Shiu discounted penalty function in the classical risk model. This is used to derive the trivariate distribution of the deficit at ruin, the surplus prior to ruin, and the surplus immediately following the second last claim before ruin. The marginal distribution of the last interclaim time before ruin is derived and studied, and its joint distribution with the claim causing ruin is derived.

Gerber–Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times

In this paper, we consider a compound Poisson risk model perturbed by a Brownian motion. We construct the bivariate cumulative distribution function of the claim size and interclaim time by Farlie–Gumbel–Morgenstern copula. The integro-differential equations and the Laplace transforms for the Gerber–Shiu functions are obtained. We also show that the Gerber–Shiu functions satisfy some defective renewal equations. For exponential claims, some explicit expressions are obtained, and numerical examples for the ruin probabilities are also given.

Extremes on the discounted aggregate claims in a time dependent risk model

This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables (rv's). Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed rv's. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims. A simulation study is performed in order to validate the results obtained in the free interest risk model.

A class of Sparre Andersen risk process

In this paper, we investigate a renewal risk model in which the distribution of the interclaim times is a mixture of two Erlang distributions. First, the Laplace transform and the defective renewal equation for the Gerber-Shiu function are derived. Then, two asymptotic results for the Laplace transform of the time of ruin are given when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively. Finally, an explicit expression for the Gerber-Shiu function is given.

Some Remarks on Delayed Renewal Risk Models

AbstractSome extensions to the delayed renewal risk models are considered. In particular, the independence assumption between the interclaim time and the subsequent claim size is relaxed, and the classical Gerber-Shiu penalty function is generalized by incorporating more variables. As a result, general structures regarding various joint densities of ruin related quantities as well as their probabilistic interpretations are provided. The numerical example in case of time-dependent claim sizes is provided, and also the usual delayed model with time-independent claim sizes is discussed including a special case with exponential claim sizes. Furthermore, asymptotic formulas for the associated compound geometric tail for the present model are derived using two alternative methods.

Constant Dividend Barrier in a Risk Model with a Generalized Farlie-Gumbel-Morgenstern Copula

In this paper, we consider the classical surplus process with a constant dividend barrier and a dependence structure between the claim amounts and the inter-claim times. We derive an integro-differential equation with boundary conditions. Its solution is expressed as the Gerber-Shiu discounted penalty function in the absence of a dividend barrier plus a linear combination of a finite number of linearly independent particular solutions to the associated homogeneous integro-differential equation. Finally, we obtain an explicit solution when the claim amounts are exponentially distributed and we investigate the effects of dependence on ruin quantities.

The Gerber–Shiu discounted penalty function in the risk process with phase-type interclaim times

In this paper, we consider the Gerber–Shiu discounted penalty function for the Sparre Anderson risk process in which the interclaim times have a phase-type distribution. By the Markov property of a joint process composed of the risk process and the underlying Markov process, we provide a new approach to prove the systems of integro-differential equations for the Gerber–Shiu functions. Closed form expressions for the Gerber–Shiu functions are obtained when the claim amount distribution is from the rational family. Finally we compute several numerical examples intended to illustrate the main results.

On a risk model with stochastic premiums income and dependence between income and loss

Labbé and Sendova (2009) [9] consider a compound Poisson risk model with stochastic premiums income. In this paper, we extend their model by assuming that there exists a specific dependence structure among the claim sizes, interclaim times and premium sizes. Assume that the distributions of the premium sizes and interclaim times are controlled by the claim sizes. When the individual premium sizes are exponentially distributed, the Laplace transforms and defective renewal equations for the (Gerber–Shiu) discounted penalty functions are obtained. When the individual premium sizes have rational Laplace transforms, we show that the Laplace transforms for the discounted penalty functions can also be obtained.

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

In this paper a discrete time renewal risk model with arbitrary interclaim times is discussed. We show that the expected discounted penalty function satisfies a recursive formula. In particular, the probability generating function of the time of ruin, as a function of the initial surplus, has a compound geometric tail. When the claim amounts follow a geometric distribution, explicit expression for the Gerber-Shiu function can be obtained for the specially chosen penalty function. The constant claim amounts and mixed geometric claim amounts are also examined.

The Gerber–Shiu penalty functions for two classes of renewal risk processes

In this paper, we study the Gerber–Shiu functions for a risk model with two independent classes of risks. We suppose that both of the two claim number processes are renewal processes with phase-type inter-claim times. By re-composing and analyzing the Markov chains associated with two given phase-type distributions, we obtain systems of integro-differential equations for two types of Gerber–Shiu functions. Explicit expressions for the Laplace transforms of the two types of Gerber–Shiu functions are established, respectively. And explicit results for the Gerber–Shiu functions are derived when the initial surplus is zero and when the two claim amount distributions are both from the rational family. Finally, an example is considered to illustrate the applicability of our main results.

Analysis of a Generalized Penalty Function in a Semi-Markovian Risk Model

In this paper an extension of the semi-Markovian risk model studied by Albrecher and Boxma (2005) is considered by allowing for general interclaim times. In such a model, we follow the ideas of Cheung et al. (2010b) and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty function, namely, the minimum surplus level before ruin and the surplus level immediately after the second last claim prior to ruin. It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation. Detailed examples are also considered when either the interclaim times or the claim sizes are exponentially distributed. Finally, we also consider the case where the claim arrival process follows a Markovian arrival process. Probabilistic arguments are used to derive the discounted joint distribution of four random variables of interest in this risk model by capitalizing on an existing connection with a particular fluid flow process.

Surplus analysis for a class of Coxian interclaim time distributions with applications to mixed Erlang claim amounts

Gerber–Shiu analysis with the generalized penalty function proposed by Cheung et al. (in press-a) is considered in the Sparre Andersen risk model with a K n family distribution for the interclaim time. A defective renewal equation and its solution for the present Gerber–Shiu function are derived, and their forms are natural for analysis which jointly involves the time of ruin and the surplus immediately prior to ruin. The results are then used to find explicit expressions for various defective joint and marginal densities, including those involving the claim causing ruin and the last interclaim time before ruin. The case with mixed Erlang claim amounts is considered in some detail.

Perturbed Risk Processes Analyzed as Fluid Flows

In this article, we present a unified phase-type approach for calculating ruin probabilities for a class of perturbed risk processes, which includes the perturbed Sparre–Andersen process with phase-type interclaim time and claim size distributions, the perturbed Markov-modulated risk process, and the MAP/PH risk process as well. The key to the solution method is the identification for a given risk process of a parallel Markov-modulated fluid flow. The resulting group of perturbed risk processes amenable to this approach we refer to as the Markov-modulated fluid-flow equivalent (MFE) class, as the ability to find an equivalent fluid flow is the only limiting factor in our analysis. The primary contributions of this work are 1) the unified, tractable, phase-type structures for the maximal aggregate loss and ladder height distributions for all variants of perturbed risk processes belonging to the MFE class, and 2) the explicit formulas for the ruin probability for all variants in this class. The full breadth of the models is illustrated through a diverse range of examples.

On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

In this paper, we consider a perturbed Sparre Andersen risk model, in which the inter-claim times are generalized Erlang(n) distributed. Under the multi-layer dividend strategy, piece-wise integro-differential equations for the discounted penalty functions are derived, and a recursive approach is applied to express the solutions. A numerical example to calculate the ruin probabilities is given to illustrate the solution procedure.

Structural properties of Gerber–Shiu functions in dependent Sparre Andersen models

The structure of various Gerber–Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the second last claim before ruin. Defective joint and marginal distributions involving these quantities are derived. Many of the properties in the Sparre Andersen model without dependence are seen to hold in the present model as well. A discussion of Lundberg’s fundamental equation and the generalized adjustment coefficient is given, and the connection to a defective renewal equation is considered. The usual Sparre Andersen model without dependence is also discussed, and in particular the case with exponential claim sizes is considered.

Dependent Risk Models with Bivariate Phase-Type Distributions

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.

Dependent Risk Models with Bivariate Phase-Type Distributions

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.