Gerber-Shiu Function Research Articles

A unifying approach to the analysis of business with random gains

In this paper, we consider a stochastic model in which a business enterprise is subject to constant rate of expenses over time and gains which are random in both time and amount. Inspired by Albrecher & Boxma (2004), it is assumed in general that the size of a given gain has an impact on the time until the next gain. Under such a model, we are interested in various quantities related to the survival of the business after default, which include: (i) the fair price of a perpetual insurance which pays the expenses whenever the available capital reaches zero; (ii) the probability of recovery by the first gain after default if money is borrowed at the time of default; and (iii) the Laplace transforms of the time of recovery and the first duration of negative capital. To this end, a function resembling the so-called Gerber–Shiu function (Gerber & Shiu (1998)) commonly used in insurance analysis is proposed. The function's general structure is studied via the use of defective renewal equations, and its applications to the evaluation of the above-mentioned quantities are illustrated. Exact solutions are derived in the independent case by assuming that either the inter-arrival times or the gains have an arbitrary distribution. A dependent example is also considered and numerical illustrations follow.

Gerber–Shiu function for the discrete inhomogeneous claim case

The discrete time risk model with non-identically distributed claims is investigated. Finite and infinite time recursive Gerber–Shiu functions are considered and the algorithm of calculation guidelines is written. Examples of ruin probability and Gerber–Shiu function behaviour are shown.

On a risk model with Markovian arrivals and tax

A risk model with Markovian arrivals and tax payments is considered. When the insurer is in a profitable situation, the insurer may pay a certain proportion of the premium income as tax payments. First, the Laplace transform of the time to cross a certain level before ruin is discussed. Second, explicit formulas for a generalized Gerber-Shiu function are established in terms of the ‘original’ Gerber-Shiu function without tax and the Laplace transform of the first passage time before ruin. Finally, the differential equations satisfied by the expected accumulated discounted tax payments until ruin are derived. An explicit expression for the discounted tax payments is also given.

A generalized penalty function for a class of discrete renewal processes

Analysis of a generalized Gerber–Shiu function is considered in a discrete-time (ordinary) Sparre Andersen renewal risk process with time-dependent claim sizes. The results are then applied to obtain ruin-related quantities under some renewal risk processes assuming specific interclaim distributions such as a discrete K n distribution and a truncated geometric distribution (i.e. compound binomial process). Furthermore, the discrete delayed renewal risk process is considered and results related to the ordinary process are derived as well.

A matrix operator approach to a risk model with two classes of claims

In this paper, we study a risk model with two independent classes of risks, in which both claim number processes are renewal processes with phasetype inter-arrival times. Using a generalized matrix Dickson-Hipp operator, a matrix Volterra integral equation for the Gerber-Shiu function is derived. And the analytical solution to the Gerber-Shiu function is also provided.

On a renewal risk process with dependence under a Farlie–Gumbel–Morgenstern copula

In this article, we consider an extension to the renewal or Sparre Andersen risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie–Gumbel–Morgenstern copula proposed by Cossette et al. (2010) for the classical compound Poisson risk model. We consider that the inter-arrival times follow the Erlang(n) distribution. By studying the roots of the generalised Lundberg equation, the Laplace transform (LT) of the expected discounted penalty function is derived and a detailed analysis of the Gerber–Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given through the compound geometric tail representation of the LT of the time to ruin. Explicit expressions for the discounted joint and marginal distribution functions of the surplus prior to the time of ruin and the deficit at the time of ruin are derived. Finally, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the LT of the time to ruin are given.

On the analysis of a general class of dependent risk processes

A generalized Sparre Andersen risk process is examined, whereby the joint distribution of the interclaim time and the ensuing claim amount is assumed to have a particular mathematical structure. This structure is present in various dependency models which have previously been proposed and analyzed. It is then shown that this structure in turn often implies particular functional forms for joint discounted densities of ruin related variables including some or all of the deficit at ruin, the surplus immediately prior to ruin, and the surplus after the second last claim. Then, employing a fairly general interclaim time structure which involves a combination of Erlang type densities, a complete identification of a generalized Gerber–Shiu function is provided. An application is given applying these results to a situation involving a mixed Erlang type of claim amount assumption. Various examples and special cases of the model are then considered, including one involving a bivariate Erlang mixture model.

The discounted penalty function with multi-layer dividend strategy in the phase-type risk model

This paper considers a Sparre Andersen model in which the inter-claim times have a phase-type distribution and the premium rate is a step function depending on the current surplus level. We derive the system of piecewise integro-differential equations for the Gerber–Shiu discounted penalty functions and obtain the closed form expressions of the Gerber–Shiu functions if the claim amount distribution belongs to the rational family. We provide a recursive approach to calculate Gerber–Shiu functions and present an example.

Non-parametric estimation of the Gerber–Shiu function for the Wiener–Poisson risk model

A non-parametric estimator of the Gerber–Shiu function is proposed for a risk process with a compound Poisson claim process plus a diffusion perturbation; the Wiener–Poisson risk model. The estimator is based on a regularized inversion of an empirical-type estimator of the Laplace transform of the Gerber–Shiu function. We show the weak consistency of the estimator in the sense of an integrated squared error with the rate of convergence.

The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model

This paper considers a Sparre Andersen model in which the inter-claim times have a phase-type distribution and the premium rate is a step function depending on the current surplus level. We derive the system of piecewise integro-differential equations for the Gerber-Shiu discounted penalty functions and obtain the closed form expressions of the Gerber-Shiu functions if the claim amount distribution belongs to the rational family. We provide a recursive approach to calculate Gerber-Shiu functions and present an example.

Numerical Method for a Markov‐Modulated Risk Model with Two‐Sided Jumps

This paper considers a perturbed Markov‐modulated risk model with two‐sided jumps, where both the upward and downward jumps follow arbitrary distribution. We first derive a system of differential equations for the Gerber‐Shiu function. Furthermore, a numerical result is given based on Chebyshev polynomial approximation. Finally, an example is provided to illustrate the method.

Asymptotic estimates of Gerber-Shiu functions in the renewal risk model with exponential claims

This paper continues to study the asymptotic behavior of Gerber-Shiu expected discounted penalty functions in the renewal risk model as the initial capital becomes large. Under the assumption that the claim-size distribution is exponential, we establish an explicit asymptotic formula. Some straightforward consequences of this formula match existing results in the field.

On the Gerber–Shiu function for a risk model with multi-layer dividend strategy

In this paper, we present a new approach to the study of the Gerber–Shiu discounted function for the risk model with multi-layer dividend strategy. The formulae for the Gerber–Shiu discounted function and ruin probability were obtained and the special case where the claim size distribution is a combination of exponentials is considered in detail.

The Gerber–Shiu function and the generalized Cramér–Lundberg model

In this paper we extend some results in Cramér [7] by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these amounts, they are interpreted either as claims made by insureds or as income from deceased annuitants, respectively. We then demonstrate that when the events’ arrival process is a renewal process, the Gerber–Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and annuity-related income has Erlang( n, β) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation.

On a Generalization of the Risk Model with Markovian Claim Arrivals

The class of risk models with Markovian arrival process (MAP) (see e.g., Neuts[ 15 ]) is generalized by allowing the waiting times between two successive events (which can be a change in the environmental state and/or a claim arrival) to have an arbitrary distribution. Using a probabilistic approach, we determine the solution for a class of Gerber–Shiu functions apart from some unknown constants when claim sizes have a mixed exponential distribution. Such constants are later determined using the more classic ruin-analytic approach. A numerical example is later considered to illustrate the tractability of the suggested methodology in the study of Gerber–Shiu functions.

Optimal dividend strategies in a Cramer–Lundberg model with capital injections and administration costs

In this paper, we consider a classical risk model with dividend payments and capital injections in the presence of both fixed and proportionals administration costs. Negative surplus or ruin is not allowed. We measure the value of a strategy by the discounted value of the dividends minus the costs. It turns out, capital injections are only made if the claim process falls below zero. Further, at the time of an injection the company may not only inject the deficit, but inject additional capital C ≥ 0 to prevent future capital injections. We derive the associated Hamilton–Jacobi–Bellman equation and show that the optimal strategy is of band type. By using Gerber–Shiu functions, we derive a method to determine numerically the solution to the integro-differential equation and the unknown value C.

On a multi-threshold compound Poisson surplus process with interest

We consider a multi-threshold compound Poisson surplus process. When the initial surplus is between any two consecutive thresholds, the insurer has the option to choose the respective premium rate and interest rate. Also, the model allows for borrowing the current amount of deficit whenever the surplus falls below zero. Starting from the integro-differential equations satisfied by the Gerber–Shiu function that appear in Yang et al. (2008), we consider exponentially and phase-type(2) distributed claim sizes, in which cases we are able to transform the integro-differential equations into ordinary differential equations. As a result, we obtain explicit expressions for the Gerber–Shiu function.

The compound Poisson risk model with dependence under a multi-layer dividend strategy

In this paper, a compound Poisson risk model with time-dependent claims is studied under a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber-Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.

On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation

In this paper, we consider a Sparre Andersen risk model where the interclaim time and claim size follow some bivariate distribution. Assuming that the risk model is also perturbed by a jump-diffusion process, we study the Gerber–Shiu functions when ruin is due to a claim or the jump-diffusion process. By using a q-potential measure, we obtain some integral equations for the Gerber–Shiu functions, from which we derive the Laplace transforms and defective renewal equations. When the joint density of the interclaim time and claim size is a finite mixture of bivariate exponentials, we obtain the explicit expressions for the Gerber–Shiu functions.

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.