Gerber-Shiu Function Research Articles

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

Randomized observation periods for compound Poisson risk model with capital injection and barrier dividend

In this paper, we model the insurance company’s surplus by a compound Poisson risk model, where the surplus process can only be observed at random observation times. It is assumed that the insurer observes its surplus level periodically to decide on dividend payments and capital injection at the interobservation time having an operatorname{Erlang}(n) distribution. If the observed surplus level is greater than zero but less than injection line b_{1} > 0, the shareholders should immediately inject a certain amount of capital to bring the surplus level back to the injection line b_{1}. If the observed surplus level is larger than dividend line b_{2} (b_{2} > b_{1}), any excess of the surplus over b_{2} is immediately paid out as dividends to the shareholders of the company. Ruin is declared when the observed surplus level is negative. We derive the explicit expressions of the Gerber–Shiu function, the expected discounted capital injection, and the expected discounted dividend payments. Numerical illustrations are also given to analyze the effect of random observation times on actuarial quantities.

The Laplace transform of ruin time with investment and barrier dividend

In order to study the Gerber-Shiu function for a compound Poisson-Geometric risk model with investment and barrier dividend, we obtain the differential-integral equation of the Gerber-Shiu function by using the method of total expectation formula. Under the assumption of exponential distribution, the explicit solution of the Laplace transform of ruin time of insurance company with investment and barrier dividend is given.

On a double barrier hybrid dividend strategy in a compound Poisson risk model with stochastic income

The paper features a hybrid dividend payment strategy on an insurance surplus with stochastic income. The hybrid dividend strategy works as follows: A delay-clock starts whenever the surplus up-shoots, due to a premium arrival, to a level between barriers ‘k’ and ‘ $$ \ell $$ ’, such that $$ 0 \leqslant u \leqslant k \leqslant \ell < \infty $$ . Further, the clock restarts along with each premium or claim arrival, provided the surplus still lies between ‘k’ and ‘ $$ \ell $$ ’. However, when the surplus exits to a level outside the range between two barriers, the clock terminates. The insurer pays a dividend of an amount exceeding the level ‘k’ to shareholders on two scenarios. In one situation, the insurer pays dividend if delay-clock alarms before a premium or claim arrival. In the other situation, the insurer pays dividend instantaneously if surplus exceeds the second barrier ‘ $$ \ell $$ ’. The aggregate premiums and claims are both following mutually independent compound Poisson processes, both having exponential amount sizes which are further independent to corresponding inter-arrival times. A scheme for determining analytic expressions for piecewise Volterra integral equations, with boundary conditions, which are satisfied by the expected discounted dividend paid before ruin and the Gerber–Shiu function are discussed in detail when the delay-clock also follows an exponential distribution. We discuss a numerical example to show the tractability of the scheme and to brief the impact of hybrid dividend strategy over the classical De Finetti barrier strategy for paying dividends.

Relations between integrated tails and moments based on the deficit at ruin in the renewal risk model

In this paper, a renewal model of risk theory is considered, and relations between tails of the equilibrium distribution of the deficit at the time of ruin and a generalization of the equilibrium distribution of non-ruin are obtained. Extensions to higher order equilibrium distribution and Gerber-Shiu functions are also provided. Distributional properties for Pareto and exponential claims are studied. The results are illustrated by numerical examples.

On the improved thinning risk model under a periodic dividend barrier strategy

&lt;abstract&gt;&lt;p&gt;In this study, we consider a periodic dividend barrier strategy in an improved thinning risk model, which indicates that insurance companies randomly receive premiums and pay dividends. In the improved model, the premium is stochastic, and the claim counting process is a p-thinning process of the premium counting process. The integral equations satisfied by the Gerber-Shiu function and the expected discounted cumulative dividend function are derived. Explicit expressions of those actuarial functions are obtained when the claim and premium sizes are exponentially distributed. We analyze and illustrate the impact of various parameters on them and obtain the optimal barrier. Finally, a conclusion is drawn.&lt;/p&gt;&lt;/abstract&gt;

On the Gerber-Shiu function of a MAP risk model with possible delayed phase-type by-claims

On the Gerber-Shiu function of a MAP risk model with possible delayed phase-type by-claims

Compound Binomial Model with Batch Markovian Arrival Process

A compound binomial model with batch Markovian arrival process was studied, and the specific definitions are introduced. We discussed the problem of ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm of the conditional finite-time nonruin probability is proposed. We also discuss research on the compound binomial model with batch Markovian arrival process and threshold dividend. Recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations were presented.

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.

Asymptotically Normal Estimators of the Gerber-Shiu Function in Classical Insurance Risk Model

Nonparametric estimation of the Gerber-Shiu function is a popular topic in insurance risk theory. Zhang and Su (2018) proposed a novel method for estimating the Gerber-Shiu function in classical insurance risk model by Laguerre series expansion based on the claim number and claim sizes of sample. However, whether the estimators are asymptotically normal or not is unknown. In this paper, we give the details to verify the asymptotic normality of these estimators and present some simulation examples to support our result.

Delayed Capital Injections for a Risk Process with Markovian Arrivals

In this paper we propose a generalisation to the Markov Arrival Process (MAP) risk model, by allowing for a delayed receipt of required capital injections whenever the surplus of an insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of a firm. We introduce a MAP risk model that allows for capital injections to be received instantaneously, or with a random delay, depending on the amount of deficit experienced by the firm. For this model, we derive a system of Fredholm integral equations of the second kind for the Gerber-Shiu function and obtain an explicit expression (in matrix form) in terms of the Gerber-Shiu function of the MAP risk model without capital injections. In addition, we show that the expected discounted accumulated capital injections and the expected discounted overall time in red, up to the time of ruin, satisfy a similar integral equation, which can also be solved explicitly. Finally, to illustrate the applicability of our results, numerical examples are given.

On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model

In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical risk model. We derive the equations and the boundary conditions satisfied by the Gerber-Shiu function, the expected discounted capital injection function and the expected discounted dividend function by assuming that the observation interval and claim amount are exponentially distributed, respectively. Numerical examples are also given to further analyze the influence of relevant parameters on the actuarial function of the risk model.

Gerber–Shiu Function in a Class of Delayed and Perturbed Risk Model with Dependence

This paper considers the risk model perturbed by a diffusion process with a time delay in the arrival of the first two claims and takes into account dependence between claim amounts and the claim inter-occurrence times. Assuming that the time arrival of the first claim follows a generalized mixed equilibrium distribution, we derive the integro-differential Equations of the Gerber–Shiu function and its defective renewal equations. For the situation where claim amounts follow exponential distribution, we provide an explicit expression of the Gerber–Shiu function. Numerical examples are provided to illustrate the ruin probability.

On Periodic Dividends for the Classical Risk Model with Debit Interest

A periodic dividend problem is studied in this paper. We assume that dividend payments are made at a sequence of Poisson arrival times, and ruin is continuously monitored. First of all, three integro-differential equations for the expected discounted dividends are obtained. Then, we investigate the explicit expressions for the expected discounted dividends, and the optimal dividend barrier is given for exponential claims. A similar study on a generalized Gerber–Shiu function involving the absolute time is also performed. To demonstrate the existing results, we give some numerical examples.

On Computations in Renewal Risk Models—Analytical and Statistical Aspects

We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show weak convergence of an approximating sequence of piecewise-deterministic Markov processes (PDMPs) for deriving the convergence of the procedures. We will use estimated PDMP characteristics in a subsequent step from simulated sample data and study its effect on the numerically computed Gerber-Shiu functions. It can be seen that the main source of instability stems from the hazard rate estimator. Interestingly, results obtained using MC methods are hardly affected by estimation.

Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability and heavy-tailed with small probability We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as an expansion with respect to powers of with known coefficients and we construct approximations from the first two terms of the aforementioned series. The main idea is based on the so-called fluid embedding that allows to put the considered risk process into the framework of spectrally negative Markov-additive processes and use its fluctuation theory developed in Ivanovs and Palmowski 2012.

Generalized expected discounted penalty function at general drawdown for Lévy risk processes

This paper considers an insurance surplus process modeled by a spectrally negative Lévy process. Instead of the time of ruin in the traditional setting, we apply the time of drawdown as the risk indicator in this paper. We study the joint distribution of the time of drawdown, the running maximum at drawdown, the last minimum before drawdown, the surplus before drawdown and the surplus at drawdown (may not be deficit in this case), which generalizes the known results on the classical expected discounted penalty function in Gerber and Shiu (1998). The results have semi-explicit expressions in terms of the q-scale functions and the Lévy measure associated with the Lévy process. As applications, the obtained result is applied to recover results in the literature and to obtain new results for the Gerber–Shiu function at ruin for risk processes embedded with a loss-carry-forward taxation system or a barrier dividend strategy. Moreover, numerical examples are provided to illustrate the results.

On the Gerber-Shiu function of a MAP risk model with possible delayed Phase-type By-claims

This paper introduces the delayed claim settlement feature in a MAP risk model. Surplus process of the model incorporates two kinds of phase-type claims, the main claim and the by-claim. The delay ...

On corrected phase-type approximations of the time value of ruin with heavy tails

Abstract We approximate Gerber–Shiu functions with heavy-tailed claims in a recently introduced risk model having both interclaim times and premiums depending on the claim sizes. We apply a technique known as “corrected phase-type approximations”. This results in adding a correction term to the Gerber–Shiu function with phase-type claim sizes. The correction term contains the heavy-tailed behavior at most once per convolution and captures the tail behavior of the true Gerber–Shiu function. We make the tail behavior specific in the classical case of one class of risk insured. After illustrating a use of such approximations, we study numerically the approximations’ relative errors for some specific penalty functions and claims distributions.

Estimating the Gerber-Shiu Function in a Compound Poisson Risk Model with Stochastic Premium Income

In this paper, we consider the compound Poisson risk model with stochastic premium income. We propose a new estimation of Gerber-Shiu function by an efficient method: Fourier-cosine series expansion. We show that the estimator is easily computed and has a fast convergence rate. Some simulation examples are illustrated to show that the estimation has a good performance when the sample size is finite.