Time-dependent Claim Research Articles

On a Perturbed Risk Model with Time-Dependent Claim Sizes

We consider a risk model perturbed by a Brownian motion, where the individual claim sizes are dependent on the inter-claim times. We study the Gerber–Shiu functions when ruin is due to a claim or the jump-diffusion process. Integro-differential equations and Laplace transforms satisfied by the Gerber–Shiu functions are obtained. Then, it is shown that the expected discounted penalty functions satisfy defective renewal equations. Explicit expressions can be obtained for exponential claim sizes. Finally, a numerical example is provided to measure the impact of the various dependence parameters in the risk model on the ruin probabilities.

On a discrete-time risk model with time-dependent claims and impulsive dividend payments

A discrete-time risk model with a mathematically tractable dependence structure between interclaim times and claim sizes is considered in the presence of an impulsive dividend strategy. Under such a strategy, once the insurer's reserve upcrosses the level b, the excess of the reserve over is paid off as dividends. We derive difference equations for both the expected discounted penalty function and the expected present value of dividend payments. Solution procedures for these difference equations are provided. When the joint distribution of the interclaim time and claim size is a finite mixture of bivariate geometric distributions, closed-form expressions are given. Numerical results for several sets of parameters are also provided to illustrate the applicability of the results obtained.

On a two-dimensional risk model with time-dependent claim sizes and risky investments

Consider a two-dimensional risk model, in which two insurance companies divide between them the claims in some specified proportions. Suppose that the claim sizes and inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure, and the surpluses of the two companies are invested into portfolios whose returns follow two different geometric Lévy processes. When the claim-size distribution is extended-regularly-varying tailed, asymptotic expressions for the ruin probability of this two-dimensional risk model are exhibited. Some numerical results are also presented to illustrate the accuracy of our asymptotic formulae.

On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays

In this paper, we consider an insurance portfolio containing several types of policies which may simultaneously face claims arising from the same catastrophe. A renewal counting process for the number of events causing claims and multivariate claim severities which are dependent on the occurrence time and/or the delay in reporting or payment are assumed. A unified model is proposed to study the time-dependent loss quantities such as the discounted aggregate reported/unreported claims and the number of the incurred but not reported (IBNR) claims. We then derive the joint moments of (i) different types of discounted aggregate claims until time t; and (ii) different types of discounted aggregate reported/unreported claims (including the total numbers of IBNR as special case) until time t. Finally, some numerical examples involving covariances and correlations of the aforementioned quantities are provided.

Ruin probability of renewal risk model with stochastic investment returns and one-sided linear time-dependent claims

This paper considers a time-dependent renewal risk model with stochastic investment returns and dependent claim sizes. In the model, the investment returns are described as a geometric Levy process, while the claim sizes are modeled by a one-sided linear process with independent and identically distributed innovations. In addition, the innovations and claim inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs, and each pair obeys a dependence structure described via the asymptotic of the conditional tail probability of the innovation given the inter-arrival time. When the innovation distribution is heavy tailed, we derive uniform estimates for ruin probabilities over certain time regions.

On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times

The structural properties of the moments of the time to ruin are studied in dependent Sparre Andersen models. The moments of the time to ruin may be viewed as generalized versions of the Gerber–Shiu function. It is shown that structural properties of the Gerber–Shiu function hold also for the moments of the time to ruin. In particular, the moments continue to satisfy defective renewal equations. These properties are discussed in detail in the model of Willmot and Woo (2012), which has Coxian interclaim times and arbitrary time-dependent claim sizes. Structural quantities needed to determine the moments of the time to ruin are specified under this model. Numerical examples illustrating the methodology are presented.

On the analysis of time dependent claims in a class of birth process claim count models

An integral representation is derived for the sum of all claims over a finite interval when the claim value depends upon its incurral time. These time dependent claims, which generalize the usual compound model for aggregate claims, have insurance applications involving models for inflation and payment delays. The number of claims process is assumed to be a (possibly delayed) nonhomogeneous birth process, which includes the Poisson process, contagion models, and the mixed Poisson process, as special cases. Known simplified compound representations in these special cases are easily generalized to the conditional case, given the number of claims at the beginning of the interval. Applications to the case involving “two stages” are also considered.

On a discrete-time risk model with general income and time-dependent claims

We consider a discrete-time risk model with general premium rate and time-dependent claim sizes, in which the interclaim time has an impact on the subsequent claim size. By studying the roots of Lundberg’s generalized equation, we first obtain an analytical expression for the generating function of the expected discounted penalty function. Then it is shown that the expected discounted penalty function satisfies a defective renewal equation. Moreover, a closed-form expression for the generating function of the time to ruin is obtained when the claim sizes have discrete Km distributions. Numerical examples are also given to illustrate the applicability of the results obtained.

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.

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.

Refinements of two-sided bounds for renewal equations

Many quantities of interest in the study of renewal processes may be expressed as the solution to a special type of integral equation known as a renewal equation. The main purpose of this paper is to provide bounds for the solution of renewal equations based on various reliability classifications. Exponential and nonexponential types of inequalities are derived. In particular, two-sided bounds with specific reliability conditions become sharp. Finally, some examples including ultimate ruin for the classical Poisson model with time-dependent claim sizes, the joint distribution of the surplus prior to and at ruin, and the excess life time, are provided.

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.

The total claims distribution under inflationary conditions

Abstract The total claims distribution over a fixed period of time with time dependent claim amounts is considered. A representation for the associated density function is found under certain conditions, including the important case with Poisson or mixed Poisson claim number processes and constant inflation. Methods of evaluation of this density are considered, and the cases with exponential claim sizes and regular variation of the tail are discussed in more detail.