Gerber-Shiu Discounted Penalty Function Research Articles

On an insurance ruin model with a causal dependence structure and perturbation

The classical compound Poisson risk model and the Sparre-Andersen risk model for insurance businesses assume that the interclaim times and the claim amounts are independently distributed. To relax the independence assumption, we consider a continuous-time risk model under which the interclaim-time distribution depends on the size of the previously occurred claim. For practical purposes, the surplus process is further assumed to be perturbed by a Brownian motion to address small financial fluctuations or investment returns. To analyze the risk associated with the event when the insurer’s ruin occurs, explicit solutions for the Gerber–Shiu discounted penalty function may be derived through the defective renewal equations provided here when claim amounts follow an arbitrary distribution. Applications with Kn family claim amounts and the Laplace transform of the ruin time are discussed in detail. An illustrative numerical example is presented to assess the impact of different perturbations on the underlying dependent-structure surplus process.

The Risk Model with a Constant Dividend Barrier Affected by a Threshold Value

This paper considers a new risk model with a constant dividend barrier, which the claim amount affected by a threshold value. The hypothesis of the model is presented and the integro-differential equation for the Gerber-Shiu penalty function is given. Then the linear solution of the Gerber-Shiu discounted penalty function is figured out. The paper also derives the integro-differential equation and the linear solution of the expected discounted dividend payments. An example is given too.

A Note on Gerber–Shiu Function with Delayed Claim Reporting under Constant Force of Interest

In this paper, we analyze the Gerber–Shiu discounted penalty function for a constant interest rate in delayed claim reporting times. Using the Poisson claim arrival scenario, we derive the differential equation of the Laplace transform of the generalized Gerber–Shiu function and show that the differential equation can be transformed to a Volterra equation of the second kind with a degenerated kernel. In the case of an exponential claim distribution, a closed-expression for the Gerber–Shiu function is obtained via sequence expansion. This result allows us to calculate the absolute (relative) ruin probability. Additionally, we discuss a method of solving the Volterra equation numerically and provide an illustration of the ruin’s probability to support the finding.

Integro-differential Equations for a Class of Delayed Renewal Risk Processes with Dependence

The Gerber-Shiu discounted penalty function is considered for a class of delayed renewal risk processes. In (Willmot 2004), special cases of the model include the stationary renewal risk model and the situation where the time until the first claim is exponentially distributed. In this paper, we consider a class of delayed and perturbed risk model with dependence between interclaim arrivals and claim sizes. The integro-differential equations for the Gerber-Shiu discounted penalty functions are derived.

Gerber-Shiu Function in a Discrete-time Risk Model with Dividend Strategy

In this paper, a discrete-time risk model with dividend strategy and a general premium rate is considered. Under such a strategy, once the insurer’s surplus hits a constant dividend barrier , dividends are paid off to shareholders at instantly. Using the roots of a generalization of Lundberg’s fundamental equation and the general theory on difference equations, two difference equations for the Gerber-Shiu discounted penalty function are derived and solved. The analytic results obtained are utilized to derive the probability of ultimate ruin when the claim sizes is a mixture of two geometric distributions. Numerical examples are also given to illustrate the applicability of the results obtained.

On a discrete interaction risk model with delayed claims and randomized dividends

In this paper, we study a discrete interaction risk model with delayed claims and randomized dividends payable at a non-negative threshold level. The recursive formula and the defective renewal equation for the Gerber-Shiu discounted penalty function are derived. Furthermore, the explicit expression for the discount-free Gerber-Shiu function is obtained. As an application, the joint distributions of the surplus immediately prior to ruin and the deficit at ruin and numerical illustration from a specific example are presented.

A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

In this paper, we propose a new generalized Gerber–Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber–Shiu discounted penalty function, we construct a relation between the original Gerber–Shiu discounted penalty function and our new generalized Gerber–Shiu discounted penalty function. Next, we use Laplace transform to derive a defective renewal equation for the generalized Gerber–Shiu discounted penalty function, and give a recursive method for solving the equation. Finally, when the claim amounts obey the exponential distribution, we give some explicit expressions for the generalized Gerber–Shiu discounted penalty function. Numerical illustrations are also given to study the effect of the parameters on the generalized Gerber–Shiu discounted penalty function.

Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model

This paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative Lévy risk model. Suppose that the claims process and the surplus process can be observed at a sequence of discrete time points. Using the observed data, the Gerber-Shiu functions are estimated by the Laguerre series expansion method. Consistent properties are studied under the large sample setting, and simulation results are also presented when the sample size is finite.

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

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 $\mathbb{E}({\mathrm{e}^{-\delta T}}{\mathbb{1}_{\{T<\infty \}}})$, where T is the time of ruin, and δ is a constant nonnegative force of interest. Theoretical results are illustrated by some numerical examples.

On a discrete interaction risk model with delayed claims and stochastic incomes under random discount rates

ABSTRACTIn this paper, we study a discrete interaction risk model with delayed claims and stochastic incomes in the framework of the compound binomial model. A generalized Gerber-Shiu discounted penalty function is proposed to analyse this risk model in which the interest rates follow a Markov chain with finite state space. We derive an explicit expression for the generating function of this Gerber-Shiu discounted penalty function. Furthermore, we derive a recursive formula and a defective renewal equation for the original Gerber-Shiu discounted penalty function. As an application, the joint distributions of the surplus one period prior to ruin and the deficit at ruin, as well as the probabilities of ruin are obtained. Finally, some numerical illustrations from a specific example are also given.

A new efficient method for estimating the Gerber–Shiu function in the classical risk model

In this paper, we propose a new efficient method for estimating the Gerber–Shiu discounted penalty function in the classical risk model. We develop the Gerber–Shiu function on the Laguerre basis, and then estimate the unknown coefficients based on sample information on claim numbers and individual claim sizes. The convergence rate of the estimate is derived. Some simulation examples are illustrated to show that the estimate performs very well when the sample size is finite. We also show that the proposed estimate outperforms other estimates in the simulation studies.

Closed-Form Absolute Ruin Problems of the Risk Models with State-Dependent Switched Claims

This letter mainly investigates a general risk model with the threshold dividend strategy under assumption that the claim amounts obey a state-dependent switched exponential distribution. By establishing the differential-integral equations for the Gerber-Shiu discounted penalty function, and applying the hypergeometric functions, the closed-form absolute ruin probability is derived.

The Gerber–Shiu discounted penalty function in the classical risk model with impulsive dividend policy

In this paper, we study the Gerber–Shiu discounted penalty function in the classical risk model with impulsive dividends. When the surplus process hits a barrier b, the dividend will be paid and the surplus is reduced to a level a. An integro-differential equation for the Gerber–Shiu discounted penalty function is derived by analyzing the evolution of the surplus process and it is solved by Dickson–Hipp operator method. For this process, we also investigate the Laplace transform of the time of ruin, the distribution of the surplus immediately before ruin and the deficit at ruin. These quantities for the special case where the claim size is exponentially distributed are obtained explicitly. Moreover, the distribution of the number of dividends is derived.

A note on compound renewal risk models with dependence

Over the last decade, there have been a significant amount of research works on compound renewal risk models with dependence. These risk models assume a dependence relation between interclaim times and claim amounts. In this paper, we pursue their investigation. We apply change of measure techniques within the compound renewal risk models with dependence to obtain exact expressions for the Gerber–Shiu discounted penalty function. We propose a more general approach than the usual one based on the random walk associated to the risk process as it is presented in the literature. More refined, our method keeps the embedded information in the sequence of claim amounts and interclaim times and enables us to derive an exact expression for the Gerber–Shiu discounted penalty function. Simulation is one of the advantages of change of measure techniques since we can find a new probability measure under which ruin occurs almost surely. In this paper, we investigate the importance sampling method based on change of measure techniques to compute several ruin measures. Numerical illustrations are carried out for specific bivariate distributions of the interclaim time and the claim amount to approximate interesting ruin measures.

Extended Gerber–Shiu functions in a risk model with interest

We consider a compound Poisson risk model with interest. The Gerber–Shiu discounted penalty function is modified with an additional penalty for reaching a level above the initial capital. We show that the problem can be split into two independent problems; an original Gerber–Shiu function and a first passage problem. We also consider the case of negative interest. Finally, we apply the results to a model considered by Embrechts and Schmidli (1994).

The Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims

A recursive formula of the Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims is obtained. In the discount-free case, an explicit formula is given. Utilizing such an explicit expression, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.

On a Risk Model With Delayed Claims Under Stochastic Interest Rates

This article studies a compound binomial risk model where the delayed claims are considered, and defines two types of individual claims, main claims and by-claims, respectively. Each by-claim is induced by the main claim and may be delayed for one time period with a certain probability. An extended definition of the Gerber-Shiu discounted penalty function is proposed to analyze this risk model in the framework of stochastic interest rates which follow a Markov chain with finite state space. By applying generating function and generalized Rouch’s theorem, we derive an explicit expression for this generalized Gerber-Shiu discounted penalty function in terms of the zeros of a determinant. Furthermore, we examine the original Gerber-Shiu discounted penalty function in the compound binomial model with delayed claims. In addition, we prove that the original Gerber-Shiu discounted penalty function satisfies a defective renewal equation, and derive the exact solution of this equation via an associated compound geometric distribution. Moreover, the closed form expressions of ruin probability and the distribution function of the surplus before ruin are obtained for some special cases. Finally, numerical results are provided to illustrate the applicability of our main result and the impact of the delay of by-claims on the Gerber-Shiu discounted penalty functions.

On the Gerber–Shiu discounted penalty function in a risk model with two types of delayed-claims and random income

This paper considers a risk model with random premium income and two types of by-claims, which is an extension to the general delayed claims process where there is only one type of by-claim and the premium income is a constant. Assume that each main claim induces one of the two by-claims, the by-claim and its associated main claim occur at the same time when the main claim amount is less than a threshold variable; otherwise, the occurrence of the by-claim will be delayed. An integral equations system for the Gerber–Shiu discounted penalty functions is presented by using auxiliary risk models. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber–Shiu penalty function is derived. By applying Rouché theorem, the Gerber–Shiu discounted penalty functions with reciprocal of the mean of premium surplus are obtained. According to Lagrange interpolation theorem, we prove that the Gerber–Shiu discounted penalty function satisfies a defective renewal equation. Finally, under the assumption of the claim sizes satisfying exponential distribution, the explicit formula of the ruin probability is derived when the discounted factor equals zero and the penalty function equals one.

On a perturbed Sparre Andersen risk model with dividend barrier and dependence

In this paper, we consider a Sparre Andersen risk model perturbed by a Brownian motion, where the inter-claim time and individual claim size follow some bivariate distribution. Assume that a barrier dividend strategy is applied to the surplus process, so that dividends are paid out whenever the surplus level attains a barrier b. Integral equations and integro-differential equations satisfied by the Gerber–Shiu discounted penalty functions and the expected discounted dividend payments are derived, and solutions are also given for some special cases.

The Gerber-Shiu Expected Penalty Function for the Risk Model with Dependence and a Constant Dividend Barrier

We consider a compound Poisson risk model with dependence and a constant dividend barrier. A dependence structure between the claim amount and the interclaim time is introduced through a Farlie-Gumbel-Morgenstern copula. An integrodifferential equation for the Gerber-Shiu discounted penalty function is derived. We also solve the integrodifferential equation and show that the solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of an associated homogeneous integrodifferential equation.