Bidimensional Risk Model Research Articles

Asymptotics for Finite-Time Ruin Probabilities of a Dependent Bidimensional Risk Model with Stochastic Return and Subexponential Claims

The paper considers a bidimensional continuous-time risk model with subexponential claims and Brownian perturbations, in which the price processes of the investment portfolio of the two lines of business are two geometric Lévy processes and the two lines of business share a common claim-number process, which is a renewal counting process. The paper mainly considers the claims of each line of business having a dependence structure. When the claims have subexponential distributions, the asymptotics of the finite-time ruin probabilities ψand(x1,x2;T) and ψsim(x1,x2;T) have been obtained. When the distributions of claims belong to the intersection of long-tailed and dominatedly varying-tailed distribution classes, the asymptotics of the finite-time ruin probability ψor(x1,x2;T) is given.

Asymptotics for ruin probabilities of a bidimensional risk model with a random number of delayed claims

This article considers a non-standard bidimensional risk model with constant force of interest, in which every main claim may cause a random number of delayed claims in an uncertain period of time. Specifically, if both the main claims and delayed claims are subexponential and follow some dependence structure, we drive precise asymptotic formulae for finite-time ruin probabilities, which generalize the results of Lu and Yuan (2022) and Li (2023b) to some extent.

Locally and globally uniform approximations for ruin probabilities of a nonstandard bidimensional risk model with subexponential claims

Locally and globally uniform approximations for ruin probabilities of a nonstandard bidimensional risk model with subexponential claims

Precise large deviations of aggregate claims in bidimensional risk model with dependence structures

In this article, we consider a bidimensional risk model with strong asymptotic independence between marginal claims having consistent varying tails and study the asymptotics of precise large deviations of aggregate claims by allowing claim-size vectors and their inter-arrival times to be arbitrarily dependent. This study can provide novel insights into the various dependence structures imposed on the involved modeling components.

Asymptotics for sum-ruin probabilities of a bidimensional risk model with heavy-tailed claims and stochastic returns

This paper considers a bidimensional risk model with geometric Lévy price processes and dependent heavy-tailed claims, where the two claim-number processes generated by the two different lines of business are almost arbitrarily dependent. When the distributions of the claims are subexponential with a positive lower Matuszewska index, an asymptotic formula for the finite-time sum-ruin probability is derived, which has a more transparent form when the distributions of the claims are regularly-varying-tailed and the two claim-number processes are homogeneous Poisson processes. Some simulation studies are conducted to verify the accuracy and sensitivity of the asymptotic result by employing the crude Monte Carlo method.

Asymptotic estimates for finite-time ruin probabilities in a generalized dependent bidimensional risk model with CMC simulations

<p style='text-indent:20px;'>This paper studies ruin probabilities of a generalized bidimensional risk model with dependent and heavy-tailed claims and additional net loss processes. When the claim sizes have long-tailed and dominated-varying-tailed distributions, precise asymptotic formulae for two kinds of finite-time ruin probabilities are derived, where the two claim-number processes from different lines of business are almost arbitrarily dependent. Under some extra conditions on the independence relation of claim inter-arrival times, the class of the claim-size distributions is extended to the subexponential distribution class. In order to verify the accuracy of the obtained theoretical result, a simulation study is performed via the crude Monte Carlo method.</p>

Asymptotic behavior for sum ruin probability of a generalized bidimensional risk model with heavy-tailed claims

This paper considers a generalized bidimensional continuous-time risk model with subexponential claims, constant force of interest and Brownian perturbations, where the claim sizes from each line of business are dependent according to some dependence structure and the two components of each claim-inter-arrival-time vector are arbitrarily dependent. Some asymptotic presentations are shown for the finite-time sum ruin probability defined as the probability that the sum of two surplus processes generated by two lines of business goes below zero over a time horizon Particularly, the claim-number processes from different lines of business can be arbitrarily dependent when the claim sizes are long-tailed and dominatedly-varying-tailed.

Precise large deviations in a bidimensional risk model with arbitrary dependence between claim-size vectors and waiting times

Consider a bidimensional risk model in which an insurer simultaneously confronts two types of claims sharing a common non-stationary arrival process, and the claim-sizes {X→k;k≥1} form a sequence of i.i.d. random vectors with nonnegative components being dependent on each other. Supposing that the univariate marginal distributions of the claim-size vectors have dominatedly varying tails, precise large deviations for the aggregate amount of claims are obtained, by allowing that the claim-size vectors and claim inter-arrival (waiting) times are arbitrarily dependent.

The infinite-time ruin probability for a bidimensional risk model with dependent geometric Lévy price processes

In this paper, we focus on a bidimensional risk model with heavy-tailed claims and geometric L?vy price processes, in which the two claim-number processes generated by the two kinds of business are not necessary to be identical and can be arbitrarily dependent. In this model, the claim size vectors (X1,Y1), (X2,Y2),... are supposed to be independent and identically distributed random vectors, but for i ? 1, each pair (Xi,Yi) follows the strongly asymptotic independence structure. Under the assumption that the claims have consistently varying tails, an asymptotic formula for the infinite-time ruin probability is established, which extends the existing results in the literature to some extent.

On asymptotic finite-time ruin probabilities of a new bidimensional risk model with constant interest force and dependent claims

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.

Asymptotic behavior for finite-time ruin probabilities in a generalized bidimensional risk model with subexponential claims

Asymptotic behavior for finite-time ruin probabilities in a generalized bidimensional risk model with subexponential claims

Uniform asymptotics for ruin probabilities of a non standard bidimensional perturbed risk model with subexponential claims

In this paper, we consider three types of finite-time ruin probabilities for a non standard bidimensional risk model perturbed by diffusion. In this model, it is assumed that the two claim-arrival processes are general counting processes and arbitrarily dependent. Moreover, the two classes of claim sizes are dependent according to a certain structure proposed in Ko and Tang (Journal of Applied Probability 45:5–95, 2008). When the claim sizes are assumed to be subexponential, we derive three uniformly asymptotic formulas for finite-time ruin probabilities over a finite interval of time horizon. The obtained results extend some existing ones in the literature.

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].

The uniform local asymptotics of the bidimensional discounted aggregate claim process

In this paper, we consider a bidimensional risk model with constant interest force, in which the claim size vector with the local subexponential marginal distributions and its inter-arrival time are subject to a local time-dependence structure.Moreover, we give some types of concrete joint distributions following the structure. Then we obtain the uniform local asymptotic expression of the discounted aggregate claim process and the total net loss process in the model.

Uniform asymptotics for the finite-time ruin probability of a generalized bidimensional risk model with Brownian perturbations

This paper considers a generalized bidimensional continuous-time risk model with heavy-tailed claims and Brownian perturbations. In this model, the claim sizes from different lines of business are tail asymptotically independent, while the claim-number processes from different lines of business are arbitrarily dependent. Under some moment conditions on the claim-number processes, an asymptotic formula is established for the finite-time ruin probability defined as the probability that the aggregate surplus process goes below zero over the time horizon . The asymptotics holds uniformly for t in some kind of interval. The moment conditions on the claim-number processes are automatically satisfied when the claim-number processes are renewal counting processes. The obtained results confirm that the Brownian perturbations have no influence on the asymptotics of the ruin probability under consideration.

Asymptotic estimates for finite-time ruin probability of a bidimensional risk model based on entrance process

Asymptotic estimates for finite-time ruin probability of a bidimensional risk model based on entrance process

A PARTICULAR BIDIMENSIONAL TIME-DEPENDENT RENEWAL RISK MODEL WITH CONSTANT INTEREST RATES

AbstractConsider a particular bidimensional risk model, in which two insurance companies divide between them in different proportions both the premium income and the aggregate claims. In practice, it can be interpreted as an insurer–reinsurer scenario, where the reinsurer takes over a proportion of the insurer's losses. Under the assumption that the claim sizes and inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure, an asymptotic expression for the ruin probability of this bidimensional risk model with constant interest rates is established.

APPROXIMATION OF THE TAIL PROBABILITIES FOR BIDIMENSIONAL RANDOMLY WEIGHTED SUMS WITH DEPENDENT COMPONENTS

AbstractLet $\left\{ {{\bi X}_k = {(X_{1,k},X_{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of independent and identically distributed random vectors whose components are allowed to be generally dependent with marginal distributions being from the class of extended regular variation, and let $\left\{ {{\brTheta} _k = {(\Theta _{1,k},\Theta _{2,k})}^{\top}, k \ge 1} \right\}$ be a sequence of nonnegative random vectors that is independent of $\left\{ {{\bi X}_k, k \ge 1} \right\}$. Under several mild assumptions, some simple asymptotic formulae of the tail probabilities for the bidimensional randomly weighted sums $\left( {\sum\nolimits_{k = 1}^n {\Theta _{1,k}} X_{1,k},\sum\nolimits_{k = 1}^n {\Theta _{2,k}} X_{2,k}} \right)^{\rm \top }$ and their maxima $({{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{1,k}} X_{1,k},{{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{2,k}} X_{2,k})^{\rm \top }$ are established. Moreover, uniformity of the estimate can be achieved under some technical moment conditions on $\left\{ {{\brTheta} _k, k \ge 1} \right\}$. Direct applications of the results to risk analysis are proposed, with two types of ruin probability for a discrete-time bidimensional risk model being evaluated.

Asymptotic ruin probabilities for a bidimensional risk model with heavy-tailed claims and non-stationary arrivals

This article studies a bidimensional risk model, in which an insurer simultaneously confronts two kinds of claims sharing a common non-stationary arrival process. Assuming that the arrival process satisfies a large deviation principle and the claim-size distributions are heavy tailed, an asymptotic formula for the corresponding ruin probability of this bidimensional risk model is obtained.

Asymptotic Ruin Probability of a Bidimensional Risk Model Based on Entrance Processes with Constant Interest Rate

In this paper, the risk model with constant interest based on an entrance process is investigated. Under the assumptions that the entrance process is a renewal process and the claims sizes satisfy a certain dependence structure, which belong to the different heavy-tailed distribution classes, the finite-time asymptotic estimate of the bidimensional risk model with constant interest force is obtained. Particularly, when inter-arrival times also satisfy a certain dependence structure, these formulas still hold.