Constant Interest Force Research Articles

Asymptotics for ruin probabilities of a bidimensional risk model with a random number of delayed claims

This article considers a non-standard bidimensional risk model with constant force of interest, in which every main claim may cause a random number of delayed claims in an uncertain period of time. Specifically, if both the main claims and delayed claims are subexponential and follow some dependence structure, we drive precise asymptotic formulae for finite-time ruin probabilities, which generalize the results of Lu and Yuan (2022) and Li (2023b) to some extent.

Uniform asymptotics for a risk model with constant force of interest and a random number of delayed claims

Consider an insurance risk model with constant force of interest under the assumption that an immediate claim is accompanied by a random number of delayed claims. In the presence of heavy tails on claim distributions and dependence structures among modelling components, we study the uniform asymptotics for an insurer's ruin-related quantities, where the uniformity holds for all time horizons varying in a relevant interval.

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.

Asymptotics for a time-dependent by-claim model with dependent subexponential claims

Consider a by-claim risk model with a constant force of interest, where each main claim may induce a by-claim after a random time. We propose a time-claim-dependent framework, that incorporates dependence between not only the waiting time and the claim but also the main claim and the corresponding by-claim. Based on this framework, we derive some asymptotic estimates for the finite-time ruin probabilities in the case of subexponential claims. We also provide examples and verify the assumptions on dependence. Numerical studies are conducted to examine the performance of these asymptotic formulas.

Finite-Time Ruin Probabilities of Bidimensional Risk Models with Correlated Brownian Motions

The present work concerns the finite-time ruin probabilities for several bidimensional risk models with constant interest force and correlated Brownian motions. Under the condition that the two Brownian motions {B1(t),t≥0} and {B2(t),t≥0} are correlated, we establish new results for the finite-time ruin probabilities. Our research enriches the development of the ruin theory with heavy tails in unidimensional risk models and the dependence theory of stochastic processes.

Asymptotics for a diffusion-perturbed risk model with dependence structures, constant interest force, and a random number of delayed claims

During extreme events such as natural or man-made disasters, it is likely that after immediate claims directly caused by the events, there are other delayed claims occurring in a certain period of future. In the classical risk model, each immediate claim is accompanied with a delayed one, but in reality each immediate claim can lead to a random number of delayed ones. To this end, we consider an insurance risk model with a random number of delayed claims as well as Brownian perturbation, dependence structures and constant interest force, and explore asymptotic estimations for an insurer’s ruin-related risks exposed to the extreme events. This article provides a novel insight to model and analyze the potential risks by fully considering the ongoing impacts of heavy-tailed claims, and thus accurately assesses the insurer’s operation risk in the long run.

Asymptotics for random-time ruin probability of a risk model with diffusion, constant interest force and non-stationary arrivals

Asymptotics for random-time ruin probability of a risk model with diffusion, constant interest force and non-stationary arrivals

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.

Asymptotic behavior for sum ruin probability of a generalized bidimensional risk model with heavy-tailed claims

This paper considers a generalized bidimensional continuous-time risk model with subexponential claims, constant force of interest and Brownian perturbations, where the claim sizes from each line of business are dependent according to some dependence structure and the two components of each claim-inter-arrival-time vector are arbitrarily dependent. Some asymptotic presentations are shown for the finite-time sum ruin probability defined as the probability that the sum of two surplus processes generated by two lines of business goes below zero over a time horizon Particularly, the claim-number processes from different lines of business can be arbitrarily dependent when the claim sizes are long-tailed and dominatedly-varying-tailed.

Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model

In this paper, we derive a closed-form expression of the tail probability of the aggregate discounted claims under homogeneous, non-homogeneous and mixed Poisson risk models with constant force of interest by using a general dependence structure between the inter-occurrence time and the claim sizes. This dependence structure is relevant since it is well known that under catastrophic or extreme events the inter-occurrence time and the claim severities are dependent.

On asymptotic finite-time ruin probabilities of a new bidimensional risk model with constant interest force and dependent claims

In this article, consider a new continuous-time bidimensional renewal risk model with constant force of interest, in which every kind of business is assumed to pay two classes of claims called the first and second ones, respectively. Suppose that the first class of claim vectors form a sequence of independent and identically distributed random vectors following a general dependence structure which share a common renewal counting process, and the second class of claim vectors, independent of the first class of claim vectors, constitute another sequence of independent and identically distributed random vectors which arrive according to two different renewal counting process. For such a model, when the claims are assumed to be subexponential or belong to the intersection of long-tailed and dominatedly varying-tailed class, some asymptotic formulas on finite-time ruin probabilities are derived. The obtained results substantially extend some existing ones in the literature.

Asymptotic sum-ruin probability for a bidimensional renewal risk model with subexponential claims

This article considers a bidimensional continuous-time renewal risk model with a constant interest force, in which the claim sizes from the same business line are dependent following a general dependence structure proposed by Ko and Tang (2008) and each pair of inter-arrival times of the two kinds of insurance claims are arbitrarily dependent. In the presence of subexponential claim sizes, the corresponding asymptotic formula for the finite-time sum-ruin probability is established.

Asymptotics for a time-dependent renewal risk model with subexponential main claims and delayed claims

This paper considers a nonstandard renewal risk model with constant force of interest, in which each main claim induces a delayed claim. Assume that the main claim sizes and the inter-arrival times obey a dependence structure, and so do the delayed claim sizes and the arrival times. Supposing that the distributions of the main and delayed claim sizes both have subexponential tails, asymptotic estimate for the finite-time ruin probability of this risk model is obtained, as the initial surplus tends to infinity.

Uniform asymptotics for the compound risk model with dependence structures and constant force of interest

Consider a nonstandard compound renewal risk model, which is more realistic in catastrophe insurance and can extend classical risk models to the case when perhaps not only one individual claim occurs in a catastrophe such as windstorm, earthquake, epidemic, strike or riot. In the presence of heavy tails and dependence structures among modelling components, we obtain the uniform asymptotics of the tail probability of discounted aggregate claims and the finite-time ruin probability for all time varying in a relevant finite or infinite interval.

Second Order Asymptotics Formula for Discounted Total Claims with Constant Interest Force

In this article, we analyse discounted total claims of insurance company, where the claim-size distribution is second order subexponential and the claim-number forms a renewal process. Using the asymptotic theory of the right-tail probability of randomly weighted sums, we study the right tail probability of discounted aggregate claims, and obtain a second order asymptotic expression, which was established uniformly for any finite-time level. The result is superior to the first order asymptotic formula.

The finite-time ruin probability of the nonhomogeneous Poisson risk model with conditionally independent subexponential claims

This paper obtains an asymptotic formula for the finite-time ruin probability of the compound nonhomogeneous Poisson risk model with a constant interest force, in which the claims are conditionally independent random variables with a common subexponential distribution. The paper also obtains some asymptotic relations of randomly weighted sums in which the weights θi are positive random variables which are bounded above and the primary random variables Xi , are conditionally independent and follow subexponential distributions.

The uniform local asymptotics of the bidimensional discounted aggregate claim process

In this paper, we consider a bidimensional risk model with constant interest force, in which the claim size vector with the local subexponential marginal distributions and its inter-arrival time are subject to a local time-dependence structure.Moreover, we give some types of concrete joint distributions following the structure. Then we obtain the uniform local asymptotic expression of the discounted aggregate claim process and the total net loss process in the model.

Asymptotic ruin probability for a by-claim risk model with pTQAI claims and constant interest force

In this paper, we consider a by-claim risk model with a constant rate of interest force, in which the main claims and the by-claims form a sequence of pTQAI nonnegative random variables and all their distributions belong to the dominatedly-varying heavy-tailed subclass. We obtain the asymptotically upper and lower bound formulas of the ultimate ruin probability for such a by-claim risk model. As its by-products, some interesting properties for pTQAI structure are also investigated. The results extend some existing ones in the literature.

Bivariate regular variation among randomly weighted sums in general insurance

The tail behavior of randomly weighted sums has become an increasingly interesting topic in applied probability and this study has played an important role in a few problems in insurance, finance, and risk management. In this paper, we extend the study to the case of non-standard bivariate regular variation and, as applications, we interpret the study in terms of bivariate processes of aggregate claims (without interest rate, with a constant force of interest, or with stochastic investment returns).

On asymptotic finite-time ruin probability of a renewal risk model with subexponential main claims and delayed claims

This paper considers a non-standard renewal risk model with constant force of interest, where each main claim may derive a delayed claim occurring after a random period of time. By means of recent asymptotic results for randomly weighted sums of subexponential random variables, we obtain some precise asymptotic expansions for the finite-time ruin probability when the main claims have a common subexponential tail.