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

of thesis entitled STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES Submitted by CHEN YIQING for the degree of Doctor of Philosophy at The University of Hong Kong in August 2009 In recent years, insurance risk studies in the presence of dependence structure and heavy-tailed claims are popular topics in actuarial science. In this thesis, we consider the renewal risk model with certain dependence structures, namely negative dependence and quasi-asymptotic independence. We also assume that claim sizes follow a heavy-tailed distribution, in particular, a subexponential distribution. We focus on studying the impact of heavy tails and dependence structures on ruin probabilities and the tail probabilities of aggregate claims. The asymptotic behavior of the ultimate ruin probability in the renewal risk model was studied in the presence of negatively dependent claim sizes and a constant force of interest. A purely probabilistic approach was employed to establish a simple asymptotic formula for the ruin probability. This result coincides with some known results in the recent literature, and discovers a phenomenon that heavy tails can offset the impact of the dependence structure of claim sizes on the ruin probability. A two-dimensional renewal risk model was considered in which the insurance company consists of two dependent classes of business. The dependence between the two classes comes from the fact that their claims arrive according to a common renewal counting process. For this model, several ruin probabilities were defined and simple asymptotic formulae for these ruin probabilities were derived. In particular, the obtained asymptotic formulae hold uniformly for all time horizons. This uniformity greatly enhances the theoretical value of the results. The next problem was to study precise estimates of the large deviation probabilities of aggregate claims with sizes which are negatively dependent. Specially, the problem for the so-called quasi renewal risk model was investigated in which inter-arrival times are also negatively dependent. A new dependence structure was considered and was termed “quasi-asymptotic independence” structure, which is more general and verifiable than some commonlyused dependence structures. The proposed dependence structure covers a wide range of positive and negative dependence. For the case of Pareto-like tails, the tail probability of the sum was proved to be asymptotically equivalent to the sum of tail probabilities. This confirms again that heavy tails offset the impact of dependence between summands. This result was then extended to randomly weighted sums and random sums. All the obtained results have obvious applications to concrete actuarial problems. In the last part of the thesis, each primary random variable representing the net loss during a period was associated with a random weight or a stochastic discount factor. Therefore, the weighted sum represents the stochastic present value of aggregate losses. Suppose that the primary random variables are long tailed, the weights are positive and bounded, and the primary random variables and their weights are independent. The tail probability of the maximum of the weighted sums was shown to be asymptotically equivalent to that of the last weighted sum. STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES

Asymptotic ruin probability of a renewal risk model with dependent by-claims and stochastic returns

ABSTRACTConsider a renewal risk model with a constant premium and a constant force of interest rate, where the claim sizes and inter-arrival times follow certain dependence structures via some restriction on their copula function. Under the assumption that the distribution of the claim-size belongs to the intersection of the class and the class , or a larger intersection class of O-subexponential distribution, class and , the infinite-time absolute ruin probabilities are derived.

Kluppelberg and Stadtmuller (1998) proved a precise asymptotic formula for the ruin probability of the classical interest force and regularly varying tailed claims when the initial capital u tends to infinity . This paper extends their results in several aspects as follows: First, the risk models are the Conditional Poisson and Non-Qici Poisson Process respectively; Second, the ruin probability is replaced by the finite time ruin probability within time T and the claimsize is of Subexponential family which is more wide then that of regularly varying's family. At last, the diffusion term of Brownian Motion is considered. construct et ψ(u;T) be the finite time ruin probability in the renewal risk model, where u is the initial capital of the company and $T$ denotes some given time bound. Under the assumption that the distribution of the claim size belongs to the Extended regular variation class, this paper obtains an asymptotic formula for ψ(u;T). This result improves the related works in the recent literature.

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.

In this paper, we consider a perturbed compound Poisson risk model with stochastic premiums and constant interest force. We obtain the upper bound and Lundberg-Cramér approximation for the infinite-time ruin probability, and consider the asymptotic formula for the finite-time ruin probability when the claim size is heavy-tailed. We show that the model in our paper has similar results to the classical risk process and some existing generalized models.

This paper studies a bidimensional renewal risk model with constant force of interest and heavy-tailed claims. We extend the existing models through allowing arbitrary dependences between each pair of inter-arrival times of the two kinds of insurance claims. A precise asymptotic formula for the finite-time ruin probability is obtained when the claims have subexponential tails. Further restricting the claim-size distributions within the class of extended regular variation, we derive the corresponding asymptotic formula for the infinite-time ruin probability.

This paper investigates the asymptotic behavior of the random-time ruin probability in a time-dependent renewal risk model with pairwise quasi-asymptotically independent and subexponential claims, where the time-dependence structure is constructed between a claim size and its inter-arrival time, and described by a conditional tail probability of the claim size given the inter-arrival time before the claim occurs. In particular, the results we obtained are also valid for the finite-time ruin probability.

Let \(R(t) = u + ct - \sum\limits_{i = 1}^{N(t)} {X_i ,t \geqslant 0} \) be the renewal risk model, with FX(x) being the distribution function of the claim amount X. Let ψ(u) be the ruin probability with initial surplus u. Under the condition of FX(x) ∈ S*(γ, γ ⩾ 0, by the geometric sum method, we derive the local asymptotic behavior for ψ(u, u + z] for every 0 < z < ∞. On one hand, the asymptotic behavior of ψ(u) can be derived from the result obtained. On the other hand, the result of this paper can be applied to the insurance risk management of an insurance company.

Research on Ruin Problems with Two-type by-claims and Delay Period Based on Compound Binomial Risk Model

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.

Uniform asymptotics for finite-time ruin probability of a bidimensional risk model

Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models

In this paper, we consider the in nite-time absolute ruin probabilities in two types of time-dependent renewal risk models with a constant premium rate and a constant interest rate. In the two models, we both as- sume that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs and that each pair obeys some dependence structure. We derive an asymptotic formula and an upper bound for the in nite-time absolute ruin probabilities in the two dependent cases, respec- tively, with the claim-size distribution belonging to the intersection of the class J ( γ ), γ > 0, and the class R -∞.

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.