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

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.

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

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.

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.

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.

Ruin Probability in a Semi-Markov Risk Model with Constant Interest Force and Heavy-Tailed Claims

In the paper we study the finite-time ruin probability in a general risk model with constant interest force, in which the claim sizes are pairwise quasi-asymptotically independent and arrive according to an arbitrary counting process, and the premium process is a general stochastic process. For the case that the claim-size distribution belongs to the consistent variation class, we obtain an asymptotic formula for the finite-time ruin probability, which holds uniformly for all time horizons varying in a relevant infinite interval. The obtained result also includes an 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.

Recently, Li [12] gave an asymptotic formula for the ultimate ruin probability in a delayed-claim risk model with constant force of interest and pairwise quasi-asymptotically independent and extended-regularly-varying-tailed claims. This paper extends Li's result to the case in which the risk model is perturbed by diffusion, the claims are consistently-varying-tailed and the main-claim interarrival times are widely lower orthant dependent.

Upper bounds for ultimate ruin probabilities in the Sparre Andersen risk model with interest and a nonlinear dividend barrier

In a recent work by Zhang et al. (J. Inequal. Appl. 2017:82, 2017) the authors studied large deviations of the aggregate amount of claims in a size-dependent renewal risk model. They claimed that they obtained a precise large-deviation formula for the case with subexponential claims and with arbitrary dependence between each claim size and its waiting time. In this short note, we point out that their main result is unfortunately not correct in general.

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.

In this paper, a dual risk model under constant force of interest is considered. The ruin probability in this model is shown to satisfy an integro-differential equation, which can then be written as an integral equation. Using the collocation method, the ruin probability can be well approximated for any gain distributions. Examples involving exponential, uniform, Pareto and discrete gains are considered. Finally, the same numerical method is applied to the Laplace transform of the time of ruin.