Renewal Risk Model Research Articles

Asymptotic sum-ruin probability for a bidimensional risk model with common shock dependence

In this paper, we consider a bidimensional continuous-time renewal risk model with common shock claim-arrival processes. When both two lines of claim sizes are assumed to be strongly subexponential, a uniformly asymptotic formula for sum-ruin probability within finite time t is established. The obtained result extends the one in Cheng and Yu [Uniform asymptotics for the ruin probabilities in a bidimensional renewal risk model with strongly subexponential claims, Stochastics 91(5) (2019), pp. 643–656].

Martingale Approach to Derive Lundberg-Type Inequalities

In this paper, we find the upper bound for the tail probability Psupn⩾0∑I=1nξI>x with random summands ξ1,ξ2,… having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ϱ1exp{−ϱ2x} with some positive constants ϱ1 and ϱ2. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.

Lundberg-type inequalities for non-homogeneous risk models

In this paper, we investigate the ruin probabilities of non-homogeneous risk models. By employing martingale method, the Lundberg-type inequalities of ruin probabilities of non-homogeneous renewal risk models are obtained under weak assumptions. In addition, for the periodic and quasi-periodic risk models the adjustment coefficients of the Lundberg-type inequalities are obtained. Finally, examples are presented to show that the estimations obtained in this paper are more accurate than those in the literatures and the ruin probabilities of non-homogeneous risk models may be fast decreasing which is impossible in the case of homogeneity.

An improved optimal trigonometric ELM algorithm for numerical solution to ruin probability of Erlang(2) risk model

In this paper, we focus on accurately calculating the numerical solution of the integral-differential equation for ruin probability in Erlang(2) renewal risk model with arbitrary claim distribution. Because the analytical solutions of the equation do not usually exist, firstly, using machine learning method in modern artificial intelligence, the activation functions in the ELM model are changed to trigonometric function, the initial conditions in the integral-differential equation are added to the ELM linear solver to get the ITELM model, and the steps and feasibility of the algorithm are strictly deduced in theory. As the analytic solution for the integral-differential equation only exists when the claim is subject to exponential distribution, and the numerical solution can be gotten with the pareto distribution. And, since the number of hidden neurons in the ITELM model is uncertain, a good numerical value of hidden neurons can only be determined through a large number of iterative tests and comparisons in the actual calculation. Then, we construct a multi-objective optimization model and algorithm, which can get the optimal number of hidden neurons to obtain the IOTELM model and algorithm. Then, in the above two cases for exponential distribution and pareto distribution, the optimal number of hidden neurons is calculated by IOTELM model and algorithm, and then corresponding ITELM models and algorithms are constructed to calculate the corresponding ruin probability. Compared with the previous numerical experiments, it can be seen that the numerical accuracy is greatly improved, which verified the versatility, feasibility and superiority of the proposed IOTELM model and algorithm.

Large deviations for sums of claims in a general renewal risk model with the regression dependent structure

The regression dependence is a practical dependent structure which provides a good mechanism to describe non-life insurance businesses, and further allows applications in various areas. In this paper, we investigate large deviations for random sums of extended negatively dependent random variables in the general renewal risk model with the regression size-dependent structure. For heavy-tailed distributions of consistently-varying tailed, we obtain the large deviation formulas.

Asymptotic finite-time ruin probabilities in a dependent bidimensional renewal risk model with subexponential claims

Asymptotic finite-time ruin probabilities in a dependent bidimensional renewal risk model with subexponential claims

On Computations in Renewal Risk Models—Analytical and Statistical Aspects

We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show weak convergence of an approximating sequence of piecewise-deterministic Markov processes (PDMPs) for deriving the convergence of the procedures. We will use estimated PDMP characteristics in a subsequent step from simulated sample data and study its effect on the numerically computed Gerber-Shiu functions. It can be seen that the main source of instability stems from the hazard rate estimator. Interestingly, results obtained using MC methods are hardly affected by estimation.

Asymptotic Behavior of Discounted Aggregate Claims in a Nonstandard Two-dimensional Risk Model with Dependent Claim Sizes

In this article, we study a nonstandard bivariate renewal risk model with dependent claim sizes, where two kinds of claim sizes constitute a sequence of independent and identically distributed random vectors satisfying a certain dependence structure, two claim-number processes are not necessarily independent. When the marginal distributions of the random vectors belong to the intersection of the class of long-tailed distributions and the class of the dominated variation distributions, we obtain joint tail asymptotic formulas of discounted total claims. The result shows that the dependence between two kinds of claims has an effect on the joint asymptotic tail probability of discounted total claims.

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.

A characterization of equivalent martingale measures in a renewal risk model with applications to premium calculation principles

Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process $S$ under a probability measure $P$ we characterize all probability measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively equivalent and $S$ remains a compound renewal process under $Q$. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.

Ruin probability in the delayed renewal risk model perturbed by a diffusion process

In this paper, we consider the risk model perturbed by an independent diffusion process with a time delay in the arrival of the first claim. We derive the distribution of the counting process of the risk process, the integro-differential equations of the ruin probabilities and generalize its defective renewal equations. With claim amounts following exponential and mixed exponential distributions, explicit expressions and asymptotic properties of the ruin probabilities are derived. Numerical illustrations of the ruin probabilities are proposed when claim amounts are exponentially and mixed exponentially distributed. We extend the results to the case of the delayed renewal risk model with exchangeable risks which captures the possible dependence between inter-arrival times of the claims and derive the associated ruin probabilities.

Precise large deviations for the aggregate claims in a dependent compound renewal risk model

We consider a dependent compound renewal risk model, where the interarrival times of accidents and the claim numbers follow a dependence structure characterized by a conditional tail probability and the claim sizes have a pairwise negatively quadrant dependence structure or a related dependence structure with the upper tail asymptotical dependence structure. When the distributions of the claim sizes belong to the dominated variation distribution class, we give the asymptotic lower and upper bounds for the precise large deviations of the aggregate claims.

Tail probability of randomly weighted sums of dependent subexponential random variables with applications to risk theory

Following the work of Cheng and Cheng (2018) [6], we reexamine the tail probability of randomly weighted sums of dependent subexponential random variables. Precisely speaking, let {Xn,n≥1} be real-valued and commonly distributed random variables satisfying a general dependence structure proposed in Ko and Tang (2008) [14], and random weights {θn,n≥1} be positive, bounded above and arbitrarily dependent random variables, but independent of {Xn,n≥1}. Under some mild conditions, we achieve the asymptotic behavior of tail probability for both randomly weighted finite and infinite sums. Finally, an application of the obtained results to a nonstandard continuous-time renewal risk model is proposed.

Second order asymptotics for ruin probabilities of the delayed renewal risk model with heavy-tailed claims

Since the pioneer work by H. Cramér, the asymptotic estimate of ruin probabilities has been extensively investigated by many researchers. Under heavy-tailed claims, Lin (2012, Insurance: Mathematics and Economics 51 (2):422–429) established the second order asymptotics for the ruin probabilities of the ordinary renewal risk model. In this paper, we extends the corresponding result in Lin (2012, Insurance: Mathematics and Economics 51 (2):422–429) to the case of the delayed renewal risk model.

An application of fractional differential equations to risk theory

This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to construct fractional integro-differential equations for the ruin probabilities in collective renewal risk models, with inter-arrival time distributions from the aforementioned family. Gamma-time risk models and fractional Poisson risk models are two specific cases among them, whose ruin probabilities have explicit solutions when claim size distributions exhibit rational Laplace transforms.

Uniform asymptotic behavior of tail probability of maxima in a time-dependent renewal risk model

In this article, we investigate the uniform asymptotic behavior for the tail probability of maxima of a random walk with and negative drift. We establish the tail probability in the frame of a time-dependent renewal risk model. In this model, we assume that the claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs. Furthermore, we suppose the tail probability distribution belongs to the strongly subexponential class.

Asymptotic ruin probabilities of a two-dimensional renewal risk model with dependent inter-arrival times

This article mainly considers the uniform asymptotics for the finite-time ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims. In this model, the two claim-number processes are arbitrarily dependent and each of them is generated by widely orthant dependent claim inter-arrival times. Two types of ruin are studied and for each type of ruin, an asymptotic formula for the finite-time ruin probability is established. These formulae possess a certain uniformity feature in the time horizon.

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.