Renewal Risk Model Research Articles

Exponential and Pareto-type bounds for the renewal function and the excess lifetime of a renewal process

In this paper, our objective is to derive two-sided bounds for the renewal function even if the interarrival times have no finite moment. By proving a relationship between the renewal function and a suitable ‘truncated' compound geometric distribution, under a truncated generalized Lundberg condition for proper renewal equations, we derive upper and lower bounds for this truncated compound geometric distribution. Using this relationship, we obtain bounds for the renewal function. Under specific truncated generalized Lundberg conditions, we obtain two-sided exponential-type bounds and Pareto-type lower bounds for the renewal function. Also, by proving a relationship between the excess lifetime of a renewal process and the df of a suitable ‘truncated deficit at ruin' of a renewal risk (or Sparre Andersen) model, we obtain two-sided exponential bounds for the df of the excess lifetime.

Moderate deviations for multidimensional aggregate claims with arbitrary dependence between claim sizes and waiting times

Consider a multidimensional renewal risk model, in which the claim sizes X k , k ≥ 1 , form a sequence of independent and identically distributed random vectors with non negative components allowed to be dependent on each other, and the inter-arrival times form a sequence of p-extended negatively dependent random variables. By allowing arbitrary dependence between claim sizes and waiting times, we obtain the moderate deviations for the multidimensional aggregate claims with consistently varying tails.

On asymptotic ruin probability for a bidimensional renewal risk model with dependent and subexponential main claims and delayed claims

On asymptotic ruin probability for a bidimensional renewal risk model with dependent and subexponential main claims and delayed claims

Joint Moments of Surplus Immediately Before Ruin and Deficit at Ruin in the Renewal Risk Model

Joint Moments of Surplus Immediately Before Ruin and Deficit at Ruin in the Renewal Risk Model

Uniform asymptotics for a renewal risk model with a random number of delayed claims

Uniform asymptotics for a renewal risk model with a random number of delayed claims

Asymptotics for a bidimensional delay-claim risk model with subexponential claims and arbitrary dependence between the generic inter-arrival time pair

This article considers a bidimensional delay-claim renewal risk model with a constant interest force, in which it is assumed that each main claim and its corresponding delayed claim from one business line satisfy a general dependence structure and the generic inter-arrival time pair from the two lines of main claims is arbitrarily dependent. In the presence of subexponential main and delayed claims, the corresponding asymptotic formulae for four types of finite-time ruin probabilities are established which extend some existing ones in the literature.

Analysis of a Dependent Perturbed Renewal Risk Model with Heavy-tailed Distributions

Analysis of a Dependent Perturbed Renewal Risk Model with Heavy-tailed Distributions

Ruin probability for renewal risk models with neutral net profit condition

In ruin theory, the net profit condition intuitively means that the sizes of the incurred random claims are on average less than the premiums gained between the successive interoccurrence times. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims’ interoccurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims are on average equal to the premiums gained between the successive interoccurrence times. We study the discrete-time risk model with N ∈ N periodically occurring independent distributions, the classical risk model, also known as the Cramér–Lundberg risk process, and the more general Sparre Andersen model.

Asymptotics for a Bidimensional Renewal Risk Model with Subexponential Main Claims and Delayed Claims

Asymptotics for a Bidimensional Renewal Risk Model with Subexponential Main Claims and Delayed Claims

On the time and aggregate claim amount until the surplus drops below zero or reaches a safety level in a jump diffusion risk model

We consider a two-barrier renewal risk model assuming that insurer's income is modeled via a Brownian motion, and the surplus is inspected only at claim arrival times. We are interested in the joint distribution of the time, number of claims and the total claim amount until the surplus process falls below zero (ruin) or reaches a safety level. We obtain a general formula for the respective joint generating function which is expressed via the distributions of the undershoot (deficit at ruin) and the overshoot (surplus exceeding safety level). We offer explicit results in the classical Poisson model, and we also study a more general renewal model assuming mixed Erlang distributed claim amounts and inter-arrival times. Our methodology is based on tilted measures and Wald's likelihood ratio identity. We finally illustrate the applicability of our theoretical results by presenting appropriate numerical examples in which we derive the distributions of interest and compare them with the ones estimated using Monte Carlo simulation.

Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims

The Parisian ruin time, which is the first time the insurer's surplus process has an excursion below level zero that exceeds a prescribed time length, has been extensively analyzed in recent years mainly in the Lévy model and its special cases. However, the cumulative Parisian ruin time, which is the first time the total time spent by the surplus process below level zero exceeds a certain time length, has been rarely considered in the literature. In this paper, we study the cumulative Parisian ruin problem in a renewal risk model with general interclaim times and exponential claims. Explicit formulas for the infinite-time cumulative Parisian ruin probability is first derived under a deterministic Parisian clock and then under an Erlang clock, where the latter case can also serve as an approximation of the former. The finite-time cumulative Parisian ruin probability is subsequently analyzed as well when the time horizon is another Erlang random variable. Our formulas are applied in various numerical examples where the interclaim times follow gamma, Weibull, or Pareto distribution. Consequently, we demonstrate that the choice of the interclaim distribution does have a significant impact on the cumulative Parisian ruin probabilities when one deviates from the exponential assumption.

Uniform Asymptotic Estimate for the Ruin Probability in a Renewal Risk Model with Cox–Ingersoll–Ross Returns

Consider an insurance risk model with arbitrary dependence structures between the claim sizes. Suppose that the risky investment in the insurer can be established by the Cox–Ingersoll–Ross model. When the claim-size distribution is heavy-tailed, a uniform asymptotic formula for ruin probability is obtained.

Uniform asymptotics for ruin probabilities of a time-dependent bidimensional renewal risk model with dependent subexponential claims

This paper considers a continuous-time bidimensional renewal risk model with subexponential claims. In this model, the claim size vectors and their inter-arrival times form a sequence of independent and identically distributed random vectors following some general dependence structures introduced by [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.]. We not only extend the results of [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.] under some weak conditions, but we also obtain an asymptotic estimate of the finite-time sum-ruin probability. Furthermore, when the distributions of claim sizes have nonzero lower Karamata indices, some explicit asymptotic formulas are established for the infinite-time ruin probabilities.

Asymptotic ruin probabilities for a renewal risk model with a random number of delayed claims

<p style='text-indent:20px;'>In this paper, we consider a non-standard renewal risk model, in which each main claim may produce a random number of delayed claims. Such a model takes full account of lasting impacts of severe claims and hence may avoid the underestimation of an insurer's operation risk within a relatively long period. We obtain asymptotic expansions for the finite-time ruin probability, given that the claim size distributions belong to the general subexponential class. When the claim size distributions are restricted to the smaller class of extended regular variation, we push our study forward to the infinite-time ruin probability.</p>

On the formulation of a stochastic model for an accumulated claim amount under renewal risk process

Traditionally an Insurance risk process is characterised by claim process using renewal process assuming claim amount is independent of inter claim time. It is usually modelled as a stochastic process such as Compound Poisson Process. It is also assumed that the premium amount is proportional to the time we refer with each claim. Depending upon the type of portfolio, the insurer can make a variety of different assumptions on the sequence of inter occurrence times and accumulated claim amount as well. In this paper we discuss a stochastic model for Renewal Risk model with different distributions to number of demands and Generalised Exponential distribution to the impact of each demand under insurance claim scenario. Assume that number of cases is independent of severity of each case throughout the model. We present the model when case frequency is Poison or Negative Binomial or Geometric and also present severity of each case with Generalised Exponential distribution.

Asymptotic ruin probabilities for a dependent renewal risk model with general investment returns and CMC simulations

Asymptotic ruin probabilities for a dependent renewal risk model with general investment returns and CMC simulations

Precise large deviations for aggregate claims of a compound renewal risk model with arbitrary dependence between claim sizes and waiting times*

Precise large deviations for aggregate claims of a compound renewal risk model with arbitrary dependence between claim sizes and waiting times*

The finite-time ruin probability of a risk model with stochastic return and subexponential claim sizes*

Consider a renewal risk model with stochastic return and stochastic perturbation, where the price process of the investment portfolio is a geometric Lévy process. When the claim sizes have a dependence structure, we derive the asymptotics of the finite-time ruin probability for all subexponential claim sizes. Particularly, when the claim sizes come from a subclass of the subexponential distribution class, the finite-time ruin probability has been estimated for claim sizes with a general dependence structure.

Uniform asymptotics for ruin probabilities of a delayed renewal risk model with one-sided linear dependence and stochastic returns

In this article, we consider a renewal risk model with by-claims, where the price process of the investment portfolio follows an exponential Lévy process. We further assume that the main claim is a one-sided linear process and there exists a certain dependence structure between the innovations and by-claims. In the presence of heavy tails, we obtain a series of uniform formulas in finite and infinite intervals. In order to better describe the obtained results, we carry on the numerical simulations.

Asymptotic analysis of a dynamic systemic risk measure in a renewal risk model

In the context of insurance, we propose a dynamic systemic risk measure based on a multi-dimensional renewal risk model to describe the instant expected shortfall of an insurer at the moment when one of its business lines suffers crisis (i.e., deficit). The asymptotic behavior of the measure is studied in both the asymptotic independence and asymptotic dependence cases, and some asymptotic formulas with the uniformity in the whole time horizon are derived when the claim size distributions belong to the class of regular variation. The obtained results do not depend on specific setups of the renewal claim-number process, and they are also insensitive to specific dependence structures in the asymptotic independence case. These facts make our results concise in form and flexible in application. More interestingly, our results for the corresponding discounted version of the measure have nothing to do with the time variable. Hence, the discounted version of the measure can be regarded approximately as a static (i.e., time-invariant) risk measure of the dynamic system.