Nonstandard Renewal Risk Research Articles

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>

Uniform asymptotics for a nonstandard compound renewal risk model with dependence structures and stochastic return on investments

Consider a nonstandard compound renewal risk model with stochastic return on investments, where the price process of investment portfolio is modeled as an exponential Lévy process. In the presence of heavy tails and dependence structures among modeling components, we study the uniform asymptotics of the tail probability of stochastic discounted aggregate claims and the finite-time ruin probability for all time varying in a relevant finite or infinite interval.

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.

A note on the uniform asymptotic behavior of the finite-time ruin probability in a nonstandard renewal risk model

Consider a nonstandard renewal risk model in which claims and interarrival times form a sequence of independent and identically distributed random pairs, with each pair obeying arbitrary dependence or size-dependence structure. In the case of heavy-tailed claims, we obtain the asymptotic behavior of finite-time ruin probability with the uniformity in time in some infinite regions.

Elementary renewal theorems for widely dependent random variables with applications to precise large deviations

On the basis of Wang and Cheng (J. Math. Anal. Appl. 384 (2011) 597–606), this paper further investigates elementary renewal theorems for counting processes generated by random walks with widely orthant dependent increments. The obtained results improve the corresponding ones of the above-mentioned paper mainly in the sense of weakening the moment conditions on the positive parts of the increments. Meanwhile, a revised version of strong law of large numbers for random walks with widely orthant dependent increments is established, which improves Theorem 1.4 of Wang and Cheng (2011) by enlarging the regions of dominating coefficients. Finally, by using the above results, some precise large deviation results for a nonstandard renewal risk model are established, in which the innovations are widely orthant dependent random variables with common heavy tails, and the inter-arrival times are also widely orthant dependent.

Large deviations for the stochastic present value of aggregate claims in the nonstandard compound renewal risk model with widely upper Orthant dependent claims

Recently, Jiang et al. (Statist. Probab. Lett. 101, 83–91) obtained the asymptotic formulas for the large deviations for the stochastic present value of aggregate claims in the renewal risk model with Pareto-type claims and stochastic return on investments, where the price process of the investment portfolio is described as a geometric Lévy process. In the paper, we extend the above results to a nonstandard compound renewal risk model with widely upper orthant dependent and dominatedly-varying-tailed claims.

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.

Uniform asymptotics for ruin probabilities in a two-dimensional nonstandard renewal risk model with stochastic returns

In this paper, we consider a two-dimensional nonstandard renewal risk model with stochastic returns, in which the two lines of claim sizes form a sequence of independent and identically distributed random vectors following a bivariate Sarmanov distribution, and the two claim-number processes satisfy a certain dependence structure. When the two marginal distributions of the claim-size vector belong to the intersection of the dominated-variation class and the class of long-tailed distributions, we obtain uniform asymptotic formulas of finite-time and infinite-time ruin probabilities.

Uniform asymptotics for ruin probabilities in a dependent renewal risk model with stochastic return on investments

In this paper, an insurer is allowed to make risk-free and risky investments, and the price process of the investment portfolio is described as an exponential Lévy process. We study the asymptotic tail behavior for a non-standard renewal risk model with dependence structures. The claim sizes are assumed to follow a one-sided linear process with independent and identically distributed step sizes, and the step sizes and inter-arrival times form a sequence of independent and identically distributed random pairs with a dependence structure. When the step-size distribution is heavy tailed, we obtain some uniform asymptotics for the finite-and infinite-time ruin probabilities.

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

Consider a nonstandard renewal risk model, in which every main claim induces a delayed by-claim. Suppose that the surplus is invested to a portfolio of one risk-free asset and one risky asset, and the main claim sizes with by-claim sizes form a sequence of pairwise quasi-asymptotically independent random variables with dominatedly varying tails. Under this setting, asymptotic behavior of the ruin probability of this renewal risk model is investigated, by establishing a weakly asymptotic formula, as the initial surplus tends to infinity. Some numerical results are also presented to illustrate the accuracy of our asymptotic formulae.

Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns

Consider a non-standard renewal risk model with dependence structures, where claim sizes follow a one-sided linear process with independent and identically distributed step sizes, the step sizes and inter-arrival times respectively form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. An insurance company is allowed to make risk-free and risky investments, where the price process of the investment portfolio follows an exponential Lévy process. When the step-size distribution is dominatedly-varying-tailed, some asymptotic estimates for the finite-and infinite-time ruin probabilities are obtained.

Optimal investment of a time-dependent renewal risk model with stochastic return

Consider an insurance company which is allowed to invest into a riskless and a risky asset under a constant mix strategy. The total claim amount is modeled by a non-standard renewal risk model with dependence between the claim size and the inter-arrival time introduced by a Farlie-Gumbel-Morgenstern copula. The price of the risky asset is described by an exponential Levy process. Based on some known results, the uniform asymptotic estimate for ruin probability with investment strategy is obtained with regularly varying tailed claims. Applying the asymptotic formula, we provide an approximation of the optimal investment strategy to maximize the expected terminal wealth subject to a risk constraint on the Value-at-Risk, which is defined with respect to finite-time discounted net loss. A numerical example is illustrated for the results, which demonstrates that big dependence parameter is advantageous for the insurer. We explain the reason by some inequalities.

Uniform Asymptotics for Discounted Aggregate Claims in Dependent Risk Models

In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Lévy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Lévy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.

Uniform Asymptotics for Discounted Aggregate Claims in Dependent Risk Models

In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Lévy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Lévy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.

Tail behavior of the sums of dependent and heavy-tailed random variables

In this paper, we investigate the tail asymptotic behavior of the partial sums, the random sums and the weighted sums of heavy-tailed random variables (r.v.s.) under two new dependence structures, respectively. The increments are real-valued and have subexponential∗ distributions, and the dependence structures can contain common linearly negatively quadrant dependent r.v.s., some positively dependent r.v.s. and some other r.v.s. The obtained results are used to derive the asymptotic estimation of the finite-time ruin probability for a nonstandard renewal risk model. In addition, some mutual relations among these two new dependence structures and some other relevant ones are discussed.

Precise Large Deviation of Claim Surplus Process in a Renewal Risk Model with Random Premium Income

In this paper, we consider a nonstandard renewal risk model in which claim sizes and corresponding inter-arrival times form a sequence of independent and identically distributed random pairs. Each pair satisfies a certain dependence structure. In addition, premium income is described by a compound Poisson process. When the distribution of claim sizes belongs to the consistent variation class, we obtain precise large deviation of claim surplus process.

Precise large deviations for compound random sums in the presence of dependence structures

In this paper, we deal with the compound random sums of dependent real-valued random variables with heavy-tailed distributions. We establish a precise large deviation result for a nonstandard renewal risk model in which innovations are extended negatively dependent real-valued random variables with a common dominatedly varying distribution function, their interarrival times are extended negatively dependent nonnegative random variables, and the numbers of innovations caused by individual events are also extended negatively dependent positive random variables. As an illustration of the obtained result, we give two applications related to some insurance risk models.

Ruin probabilities with pairwise quasi-asymptotically independent and dominatedly-varying tailed claims

This paper considers the nonstandard renewal risk model in which a part of surplus is invested into a Black-Scholes market whose price process is modelled by a geometric Brownian motion, claim sizes form a sequence of not necessarily identically distributed and pairwise quasi-asymptotically independent random variables with dominatedly-varying tails. The authors obtain a weakly asymptotic formula for the finite-time and infinite-time ruin probabilities. In particular, if the claims are identically distributed and consistently-varying tailed, then an asymptotic formula is presented.

Asymptotics in a time-dependent renewal risk model with stochastic return

In this paper we study the asymptotic tail behavior for a non-standard renewal risk model with a dependence structure and stochastic return. An insurance company is allowed to invest in financial assets such as risk-free bonds and risky stocks, and the price process of its portfolio is described by a geometric Lévy process. By restricting the claim-size distribution to the class of extended regular variation (ERV) and imposing a constraint on the Lévy process in terms of its Laplace exponent, we obtain for the tail probability of the stochastic present value of aggregate claims a precise asymptotic formula, which holds uniformly for all time horizons. We further prove that the corresponding ruin probability also satisfies the same asymptotic formula.