Subexponential Claims Research Articles

Two-dimensional ruin probability for subexponential claim size

TWO-DIMENSIONAL RUIN PROBABILITY FOR SUBEXPONENTIAL CLAIM SIZEWe analyse the asymptotics of ruin probabilities of two insurance companies or two branches of the same company that divide between them both claims and premiums in some specified proportions when the initial reserves of both companies tend to infinity, and generic claim size is subexponential.

A revisit to asymptotic ruin probabilities for a bidimensional renewal risk model

Recently, Yang and Li (2014) studied a bidimensional renewal risk model with constant force of interest and dependent subexponential claims. Under the special Farlie–Gumbel–Morgenstern dependence structure and a technical moment condition on the claim-number process, they derived an asymptotic expansion for the finite-time ruin probability. In this paper, we show that their result can be extended to a much more general dependence structure without any extra condition on the renewal claim-number process. We also give some asymptotic expansions for the corresponding infinite-time ruin probability within the scope of extended regular variation.

Some asymptotic results of the ruin probabilities in a two-dimensional renewal risk model with some strongly subexponential claims

In this paper, a two-dimensional renewal risk model of insurance business with some strongly subexponential claim sizes is considered. Three types of the finite-time ruin probabilities under this model are discussed. We obtain the asymptotic upper and lower bounds for one type, and the asymptotic formulas for the others, which hold uniformly in a corresponding region, respectively.

Precise large deviation results for sums of sub-exponential claims in a size-dependent renewal risk model

This paper deals with a size-dependent renewal risk model in which claim sizes and inter-occurrence times correspondingly form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure described via the conditional distribution of the inter-occurrence time given the subsequent claim size being large. The impact of this dependence structure on the tail behavior of aggregated claims is investigated and then a precise large deviation formula for the aggregate amount of sub-exponential claims is obtained.

The finite-time ruin probability for an inhomogeneous renewal risk model

In the paper, we give an asymptotic formula for the finite-time ruin probability in a generalized renewal risk model. We consider the renewal risk model with independent strongly subexponential claim sizes and independent not necessarily identically distributed inter occurrence times having finite variances. We find out that the asymptotic formula for the finite-time ruin probability is insensitive to the homogeneity of inter-occurrence times.

Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims

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.

Uniform asymptotics for random time ruin probability with subexponential claims and constant interest rate

Consider the probability of random time ruin in the renewal risk model with the general nonnegative and non decreasing premium process and constant interest rate. We obtain a uniform asymptotic formula for random time τ and subexponential distribution.

Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims

This paper considers a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution, we derive for the finite-time ruin probability an explicit asymptotic formula.

Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims

In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give three examples of non-stationary and non-renewal point processes: Hawkes process, Cox process with shot noise intensity and self-correcting point process. We also show some aggregate claims results for these three examples.

Uniform asymptotics of the finite-time ruin probability for all times

In this paper, we study the asymptotics of the finite-time ruin probability for a generalized renewal risk model with independent strong subexponential claim sizes and widely lower orthant dependent inter-occurrence times. This widely lower orthant dependence structure can include some common negative dependent random variables (r.v.s) such as negatively lower orthant dependent r.v.s, some positive dependent r.v.s and some other dependent r.v.s. For this generalized renewal risk model, we show that the asymptotics of the finite-time ruin probability hold uniformly for all times in the sense that the magnitude of the initial reserve of some insurance portfolio does not influence the length of the operating time. Further, the uniform asymptotics of the random-time ruin probability in the above model have been obtained.

The finite-time ruin probability in two non-standard renewal risk models with constant interest rate and dependent subexponential claims

This paper considers an ordinary renewal risk model and a compound renewal risk model with constant interest rate, subexponential claims and a general premium process. We derive some asymptotic results on the finite-time ruin probabilities.

Insurance risk capital for the Sparre Andersen model with geometric Lévy stochastic returns

Some multi-period insurance risk economic capital models that include the effects of heavy-tail claims and random returns are considered. They are based on the Sparre Andersen risk model with geometric Levy stochastic returns. The random accumulated surplus over an arbitrary finite time horizon is decomposed into insurance risk, market risk and future profit components. A protection against the solvency risk of the policyholders is obtained by applying the VaR (CVaR) measure to the insurance risk component and defines a multi-period insurance risk VaR (CVaR) economic capital. A classical asymptotic result by Resnick and Willekens [28] on the tail probability of moving averages with random coefficients is applied to the accumulated aggregate claims random variable for claim size distributions with regularly varying tail to derive asymptotic formulas for these multi-period insurance risk economic capitals. Numerical examples with a Pareto claim size distribution reveal interesting features and differences between these two solvency rules. Since the preceding results exclude the log-normal and the heavy-tailed Weibull claim size distributions, we consider also an extension to sub-exponential claim sizes for the compound Poisson model with constant force of interest, which is based on Hao and Tang [12]. The obtained results are compared with the standard Solvency II specification of the non-life insurance risk.

Lower and upper bounds of large deviation for sums of subexponential claims in a multi-risk model

In view of the actual condition of the insurance company, a multi-risk model is proposed. The lower and upper bounds for the sums of subexponential claims in this model are given. The proof method is based on the results of the total claim amount under subexponential class.

Asymptotic behavior of the Gerber–Shiu discounted penalty function in the Erlang(2) risk process with subexponential claims

We investigate the asymptotic behavior of the Gerber–Shiu discounted penalty function ɸ(u) = E(e−δT 1{T <∞} | U(0) = u), where T denotes the time to ruin in the Erlang(2) risk process. We obtain an asymptotic expression for the discounted penalty function when claim sizes are subexponentially distributed.

Higher-order expansions for compound distributions and ruin probabilities with subexponential claims

Let X i (i=1,2, …) be a sequence of subexponential positive independent and identically distributed random variables. In this paper, we offer two alternative approaches to obtain higher-order expansions of the tail of and subsequently for ruin probabilities in renewal risk models with claim sizes X i . In particular, these emphasize the importance of the term for the accuracy of the resulting asymptotic expansion of . Furthermore, we present a more rigorous approach to the often suggested technique of using approximations with shifted arguments. The cases of a Pareto type, Weibull and Lognormal distribution for X i are discussed in more detail and numerical investigations of the increase in accuracy by including higher-order terms in the approximation of ruin probabilities for finite realistic ranges of s are given.

Asymptotic behaviour of the finite‐time ruin probability in renewal risk models

AbstractIn this paper we study the tail behaviour of the probability of ruin within finite time t, as initial risk reserve x tends to infinity, for the renewal risk model with strongly subexponential claim sizes. The asymptotic formula holds uniformly for t∈[f(x), ∞), where f(x) is an infinitely increasing function, and substantially extends the result of Tang (Stoch. Models 2004; 20:281–297) obtained for the class of claim distributions with consistently varying tails. Two examples illustrate the result. Copyright © 2008 John Wiley & Sons, Ltd.

Precise large deviation results for the total claim amount under subexponential claim sizes

The paper deals with the renewal risk model. A precise large deviation result in the case of subexponential claim sizes is proved. As a special case, the example of Pareto distributed claim sizes and inter-occurrence times is investigated.

Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes

The paper deals with the Sparre Andersen risk model. We study the tail behaviour of the finite-time ruin probability, Ψ ( x , t ) , in the case of subexponential claim sizes as initial risk reserve x tends to infinity. The asymptotic formula holds uniformly for t in a corresponding region and reestablishes a formula of Tang [Tang, Q., 2004a. Asymptotics for the finite time ruin probability in the renewal model with consistent variation. Stochastic Models 20, 281–297] obtained for the class of claim distributions having consistent variation.

The finite-time ruin probability of the compound Poisson model with constant interest force

In this paper, we establish a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and subexponential claims in the case that the initial surplus is large. The formula is consistent with known results for the ultimate ruin probability and, in particular, is uniform for all time horizons when the claim size distribution is regularly varying tailed.

The finite-time ruin probability of the compound Poisson model with constant interest force

In this paper, we establish a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and subexponential claims in the case that the initial surplus is large. The formula is consistent with known results for the ultimate ruin probability and, in particular, is uniform for all time horizons when the claim size distribution is regularly varying tailed.