Finite-time Ruin Probability 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.

Finite-time ruin probability of a risk model with perturbation and subexponential main claims and by-claims

The paper considers a nonstandard risk model with stochastic return and perturbation, in which the price process of the investment portfolio is described as a geometric Lévy process and each main claim may induce a delayed by-claim. When the main claims and by-claims have subexponential distributions, we obtain some asymptotic estimations of the finite-time ruin probability.

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.

Uniform Asymptotics for Finite-Time Ruin Probabilities of Risk Models with Non-Stationary Arrivals and Strongly Subexponential Claim Sizes

This paper considers the one- and two-dimensional risk models with a non-stationary claim-number process. Under the assumption that the claim-number process satisfies the large deviations principle, the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model are obtained for the strongly subexponential claim sizes. Further, as an application of the result of one-dimensional risk model, we derive the uniform asymptotics for a kind of finite-time ruin probability in a two dimensional risk model sharing a common claim-number process which satisfies the large deviations principle.

The ruin probability of a discrete risk model with unilateral linear dependent claims

<abstract><p>This article focuses on analyzing the finite-time ruin probability within a specific class of discrete risk models. These models incorporate dependent claims, an interest rate component, and stationary noise terms exhibiting semi-heavy-tailed behavior. In this framework, the claim amount follows a unilateral linear dependent process with independent and identically distributed noise terms, while the discount factor is determined by both the interest rate and time. The finite-time ruin probability has been derived under insurance risk conditions resembling the gamma distribution.</p></abstract>

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

This article considers a bidimensional delay-claim renewal risk model with a constant interest force, in which it is assumed that each main claim and its corresponding delayed claim from one business line satisfy a general dependence structure and the generic inter-arrival time pair from the two lines of main claims is arbitrarily dependent. In the presence of subexponential main and delayed claims, the corresponding asymptotic formulae for four types of finite-time ruin probabilities are established which extend some existing ones in the literature.

Uniform asymptotics for finite-time ruin probabilities of a bidimensional compound risk model with stochastic returns

Consider a continuous-time bidimensional compound risk model with stochastic returns, where an insurance company operates two lines of business at the same time and is allowed to invest its wealth into financial assets. In this model, each accident may cause a random number of heavy-tailed claims and the claim sizes from the same line of business are upper tail asymptotically independent, while the accident arrival processes from different lines of business are arbitrarily dependent. Under some moment conditions on the accident arrival processes, some uniform asymptotic formulae for finite-time ruin probabilities are established.

On a time-changed variant of the generalized counting process

AbstractIn this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

Ruin in a continuous-time risk model with arbitrarily dependent insurance and financial risks triggered by systematic factors

This paper is devoted to asymptotic analysis for a continuous-time risk model with the insurance surplus process and the log-price process of the investment driven by two dependent jump-diffusion processes. We take into account arbitrary dependence between the insurance claims and their corresponding investment return jumps caused by a sequence of systematic factors, whose arrival times constitute a renewal counting process. Under the framework of regular variation, we obtain a simple and unified asymptotic formula for the finite-time ruin probability as the initial wealth becomes large. It turns out that, in the weakly dependent case, the tails of the claims determine the exact decay rate of the finite-time ruin probability while the investment return jumps only contribute to the coefficient of the asymptotic formula; however, in the strongly dependent case, they both produce essential impacts on the finite-time ruin probability which is under-estimated in the weakly dependent case.

Asymptotics for a time-dependent by-claim model with dependent subexponential claims

Consider a by-claim risk model with a constant force of interest, where each main claim may induce a by-claim after a random time. We propose a time-claim-dependent framework, that incorporates dependence between not only the waiting time and the claim but also the main claim and the corresponding by-claim. Based on this framework, we derive some asymptotic estimates for the finite-time ruin probabilities in the case of subexponential claims. We also provide examples and verify the assumptions on dependence. Numerical studies are conducted to examine the performance of these asymptotic formulas.

Finite-Time Ruin Probabilities of Bidimensional Risk Models with Correlated Brownian Motions

The present work concerns the finite-time ruin probabilities for several bidimensional risk models with constant interest force and correlated Brownian motions. Under the condition that the two Brownian motions {B1(t),t≥0} and {B2(t),t≥0} are correlated, we establish new results for the finite-time ruin probabilities. Our research enriches the development of the ruin theory with heavy tails in unidimensional risk models and the dependence theory of stochastic processes.

Ruin Analysis on a New Risk Model with Stochastic Premiums and Dependence Based on Time Series for Count Random Variables

In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among the claim numbers of consecutive periods is described by the integer-valued moving average (INMA(1)) process. To measure the risk of the model quantitatively, we study the explicit expression for a function whose solution is defined as the Lundberg adjustment coefficient and give the Lundberg approximation formula for the infinite-time ruin probability. In the case of heavy-tailed claim sizes, we establish the asymptotic formula for the finite-time ruin probability via the large deviations of the aggregate claims. Two numerical examples are provided in order to illustrate our theoretical findings.

Uniform asymptotics for ruin probabilities of a time-dependent bidimensional renewal risk model with dependent subexponential claims

This paper considers a continuous-time bidimensional renewal risk model with subexponential claims. In this model, the claim size vectors and their inter-arrival times form a sequence of independent and identically distributed random vectors following some general dependence structures introduced by [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.]. We not only extend the results of [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.] under some weak conditions, but we also obtain an asymptotic estimate of the finite-time sum-ruin probability. Furthermore, when the distributions of claim sizes have nonzero lower Karamata indices, some explicit asymptotic formulas are established for the infinite-time ruin probabilities.

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 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>

The finite-time ruin probability of a risk model with stochastic return and subexponential claim sizes*

Consider a renewal risk model with stochastic return and stochastic perturbation, where the price process of the investment portfolio is a geometric Lévy process. When the claim sizes have a dependence structure, we derive the asymptotics of the finite-time ruin probability for all subexponential claim sizes. Particularly, when the claim sizes come from a subclass of the subexponential distribution class, the finite-time ruin probability has been estimated for claim sizes with a general dependence structure.

Finite-time ruin probabilities using bivariate Laguerre series

In this paper, we revisit the finite-time ruin probability in the classical compound Poisson risk model. Traditional general solutions to finite-time ruin problems are usually expressed in terms of infinite sums involving the convolutions related to the claim size distribution and their integrals, which can typically be evaluated only in special cases where the claims follow exponential or (more generally) mixed Erlang distribution. We propose to tackle the partial integro-differential equation satisfied by the finite-time ruin probability and develop a new approach to obtain a solution in terms of bivariate Laguerre series as a function of the initial surplus level and the time horizon for a large class of light-tailed claim distributions. To illustrate the versatility and accuracy of our proposed method which is easy to implement, numerical examples are provided for claim amount distributions such as generalized inverse Gaussian, Weibull and truncated normal where closed-form convolutions are not available in the literature.

On the evaluation of ruin probabilities in a generalized dual binomial risk model using Markov property

This article is concerned with a reasonably generalized dual Binomial risk model where the periodic premium is one and in each period there is at most one claim and non-negative claim amounts are independent and identically distributed random variables. Our main purpose is to evaluate the probability of time to ruin and ruin probabilities. First, we present a formula for probability of time to ruin based on the convolution of claim amount. We derive a recursive expression for the finite time ruin probability based on the probability of time to ruin and classical ruin probability using the Markov property and law of total probability when surplus process stays below or at zero at least for some fixed duration of time. Moreover, we obtain an explicit expression for the corresponding infinite time ruin probability as a limiting case. Finally, to illustrate our method we calculate the numerical values of these probabilities for some different types of models of probability mass functions for claim amounts.

Diffusion Approximations of the Ruin Probability for the Insurer–Reinsurer Model Driven by a Renewal Process

We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume that the flow of claims is governed by a general renewal process. A simple ruin probability formula for the model is known only in infinite time for the special case of the Poisson process and exponentially distributed claims. Therefore, there is a need for simple analytical approximations. In the literature, in the infinite-time case, for the Poisson process, a De Vylder-type approximation has already been introduced. The idea of the diffusion approximation presented here is based on the weak convergence of stochastic processes, which enables one to replace the original risk process with a Brownian motion with drift. By applying this idea to the insurer–reinsurer model, we obtain simple ruin probability approximations for both finite and infinite time. We check the usefulness of the approximations by studying several claim amount distributions and comparing the results with the De Vylder-type approximation and Monte Carlo simulations. All the results show that the proposed approximations are promising and often yield small relative errors.

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.