A risk model with renewal shot-noise Cox process

In this paper we generalise the risk models beyond the ordinary framework of affine processes or Markov processes and study a risk process where the claim arrivals are driven by a Cox process with renewal shot-noise intensity. The upper bounds of the finite-horizon and infinite-horizon ruin probabilities are investigated and an efficient and exact Monte Carlo simulation algorithm for this new process is developed. A more efficient estimation method for the infinite-horizon ruin probability based on importance sampling via a suitable change of probability measure is also provided; illustrative numerical examples are also provided.     

Finite time ruin probabilities for tempered stable insurance risk processes

We study the probability of ruin before time t$t$ for the family of tempered stable Lévy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations based on importance sampling using stable processes. Theoretical consequences of the asymptotic formulae indicate that some care is needed in the choice of parameters to avoid exponential growth (in time) of the ruin probabilities in these models. This, in particular, applies to the inverse Gaussian process when the safety loading is less than one.     

The utility of deductibles from the insurer's point of view

The relationship between the premiums for deductible cover and for full cover are analyzed with respect to the utility for the insurer. Theorem 1 shows that within natural bounds for the premiums neither form of coverage is generally preferred by all insurers. Theorems 2–4 indicate that in many realistic cases a deductible cover requires a higher risk loading factor by the insurer than a full insurance cover. Theorem 2(b) also contains an exception to this rule.     

Risk Processes with Non-stationary Hawkes Claims Arrivals

We consider risk processes with non-stationary Hawkes claims arrivals, and we study the asymptotic behavior of infinite and finite horizon ruin probabilities under light-tailed conditions on the claims. Moreover, we provide asymptotically efficient simulation laws for ruin probabilities and we give numerical illustrations of the theoretical results.     

Some comparison results for finite-time ruin probabilities in the classical risk model

This paper aims at showing how an ordering on claim amounts can influence finite-time ruin probabilities. Until now such a question was examined essentially for ultimate ruin probabilities. Over a finite horizon, a general approach does not seem possible but the study is conducted under different sets of conditions. This primarily covers the cases where the initial reserve is null or large.     

Sensitivity analysis for ruin probabilities: canonical risk model

The surplus process of an insurance portfolio is defined as the wealth obtained by the premium payments minus the reimbursements made at the time of claims. When this process becomes negative (if ever), we say that ruin has occurred. The general setting is the Gambler's Ruin Problem. In this paper we address the problem of estimating derivatives (sensitivities) of ruin probabilities with respect to the rate of accidents. Estimating probabilities of rare events is a challenging problem, since naïve estimation is not applicable. Solution approaches are very recent, mostly through the use of importance sampling techniques. Sensitivity estimation is an even harder problem for these situations. We shall study three methods for estimating ruin probabilities: one via importance sampling (IS), and two others via indirect simulation: the storage process (SP), which restates the problems in terms of a queuing system, and the convolution formula (CF). To estimate the sensitivities, we apply the Rare Perturbation Analysis (RPA) method to IS, the Infinitesimal Perturbation Analysis (IPA) method to SP and the score function method to CF. Simulation methods are compared in terms of their efficiency, a criterion that appropriately weighs precision and CPU time. As well, we indicate how other criteria such as set-up time and prior formulae development may actually be problem-dependent.     

An asymptotic relationship for ruin probabilities under heavy-tailed claims

The famous Embrechts-Goldie-Veraverbeke formula shows that, in the classical Cramér-Lundberg risk model, the ruin probabilities satisfy \(R(x, \infty ) \sim \rho ^{ - 1} \bar F_e (x)\) if the claim sizes are heavy-tailed, where F_e denotes the equilibrium distribution of the common d.f. F of the i.i.d. claims, ? is the safety loading coefficient of the model and the limit process is for x → ∞. In this paper we obtain a related local asymptotic relationship for the ruin probabilities. In doing this we establish two lemmas regarding the n-fold convolution of subexponential equilibrium distributions, which are of significance on their own right.     

Ruin probabilities in the discrete time renewal risk model

In this paper, we study the discrete time renewal risk model, an extension to Gerber’s compound binomial model. Under the framework of this extension, we study the aggregate claim amount process and both finite-time and infinite-time ruin probabilities. For completeness, we derive an upper bound and an asymptotic expression for the infinite-time ruin probabilities in this risk model. Also, we demonstrate that the proposed extension can be used to approximate the continuous time renewal risk model (also known as the Sparre Andersen risk model) as Gerber’s compound binomial model has been proposed as a discrete-time version of the classical compound Poisson risk model. This allows us to derive both numerical upper and lower bounds for the infinite-time ruin probabilities defined in the continuous time risk model from their equivalents under the discrete time renewal risk model. Finally, the numerical algorithm proposed to compute infinite-time ruin probabilities in the discrete time renewal risk model is also applied in some of its extensions.     

On the Markov-modulated insurance risk model with tax

In this paper, we consider the Markov-modulated insurance risk model with tax. We assume that the claim inter-arrivals, claim sizes and premium process are influenced by an external Markovian environment process. The considered tax rule, which is the same as the one considered by Albrecher and Hipp [Blätter DGVFM 28(1):13–28, 2007], is to pay a certain proportion of the premium income, whenever the insurer is in a profitable situation. A system of differential equations of the non-ruin probabilities, given the initial environment state, are established in terms of the ruin probabilities under the Markov-modulated insurance risk model without tax. Furthermore, given the initial state, the differential equations satisfied by the expected accumulated discounted tax until ruin are also derived. We also give the analytical expressions for them by iteration methods.     

Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin

We consider the classical risk model and carry out a sensitivity and robustness analysis of finite-time ruin probabilities. We provide algorithms to compute the related influence functions. We also prove the weak convergence of a sequence of empirical finite-time ruin probabilities starting from zero initial reserve toward a Gaussian random variable. We define the concepts of reliable finite-time ruin probability as a Value-at-Risk of the estimator of the finite-time ruin probability. To control this robust risk measure, an additional initial reserve is needed and called Estimation Risk Solvency Margin (ERSM). We apply our results to show how portfolio experience could be rewarded by cut-offs in solvency capital requirements. An application to catastrophe contamination and numerical examples are also developed.     

On the decomposition of the absolute ruin probability in a perturbed compound Poisson surplus process with debit interest

We consider a compound Poisson surplus process perturbed by diffusion with debit interest. When the surplus is below zero or the company is on deficit, the company is allowed to borrow money at a debit interest rate to continue its business as long as its debt is at a reasonable level. When the surplus of a company is below a certain critical level, the company is no longer profitable, we say that absolute ruin occurs at this situation. In this risk model, absolute ruin may be caused by a claim or by oscillation. Thus, the absolute ruin probability in the model is decomposed as the sum of two absolute ruin probabilities, where one is the probability that absolute ruin is caused by a claim and the other is the probability that absolute ruin is caused by oscillation. In this paper, we first give the integro-differential equations satisfied by the absolute ruin probabilities and then derive the defective renewal equations for the absolute ruin probabilities. Using these defective renewal equations, we derive the asymptotical forms of the absolute ruin probabilities when the distributions of claim sizes are heavy-tailed and light-tailed. Finally, we derive explicit expressions for the absolute ruin probabilities when claim sizes are exponentially distributed.     

Risk processes with shot noise Cox claim number process and reserve dependent premium rate

We consider a suitable scaling, called the slow Markov walk limit, for a risk process with shot noise Cox claim number process and reserve dependent premium rate. We provide large deviation estimates for the ruin probability. Furthermore, we find an asymptotically efficient law for the simulation of the ruin probability using importance sampling. Finally, we present asymptotic bounds for ruin probabilities in the Bayesian setting.     

具有Vasicek利率的风险过程的破产概率的分解(英文)

In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities： the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.     

一类基于进入过程保险风险模型的破产概率(英文)

In this note,one kind of insurance risk models with the policies having multiple validity times are investigated.Explicit expressions for the ruin probabilities are obtained by using the martingale method.As a consequence,the obtained probability serves as an upper bound for the ruin probability of a newly developed entrance processes based risk model.     

Bayesian estimation of finite time ruin probabilities

In this paper, we consider Bayesian inference and estimation of finite time ruin probabilities for the Sparre Andersen risk model. The dense family of Coxian distributions is considered for the approximation of both the inter‐claim time and claim size distributions. We illustrate that the Coxian model can be well fitted to real, long‐tailed claims data and that this compares well with the generalized Pareto model. The main advantage of using the Coxian model for inter‐claim times and claim sizes is that it is possible to compute finite time ruin probabilities making use of recent results from queueing theory. In practice, finite time ruin probabilities are much more useful than infinite time ruin probabilities as insurance companies are usually interested in predictions for short periods of future time and not just in the limit. We show how to obtain predictive distributions of these finite time ruin probabilities, which are more informative than simple point estimations and take account of model and parameter uncertainty. We illustrate the procedure with simulated data and the well‐known Danish fire loss data set. Copyright © 2009 John Wiley & Sons, Ltd.     

一类相依两险种风险模型的分类破产概率

本文考虑文[1]中引入的一类索赔达到计数过程相关的两险种风险模型.利用更新方法,获得了该风险模型的分类破产概率的渐进结果,并给出了指数索赔情形下分类破产概率的表达式,从而改进了文[1]中的相关结果.     

投资收益下的两类双二项风险模型的破产概率及比较

本文讨论了含投资因素的双二项风险模型,得到了破产概率表达式,并对几类相关的双二项风险模型的调节系数及破产概率上界进行了比较.     

Multirisks Model and Finite-Time Ruin Probabilities

A multirisks model is constructed that describes the evolution in discrete-time of an insurance portfolio covering several interdependent risks. The main problem under study is the determination of the probabilities of ruin over a finite horizon, for one or more risks. An underlying polynomial structure in the expression of these probabilities is exhibited. This result is then used to provide a simple recursive method for their numerical evaluation. Furthermore, it is shown qualitatively that a stronger positive-type dependence between the risks increases the non-ruin probabilities. Some illustrations enhance the efficiency, in time and precision, of the developed algorithm.     

Multivariate risk models under heavy‐tailed risks

In this paper, we consider four common types of ruin probabilities for a discrete‐time multivariate risk model, where the insurer is assumed to be exposed to a vector of net losses resulting from a number of business lines over each period. By assuming a large initial capital for the risk model and regularly varying distributions for the net losses, we establish some interesting asymptotic estimates for ruin probabilities in terms of the upper tail dependence function of the net loss vector. Our results insightfully characterize how the dependence structure among the individual net losses affect the ruin probabilities in an asymptotic sense, and more importantly, from our main results, explicit asymptotic estimates for those ruin probabilities can be obtained via specifying a copula for the net loss vectors. Copyright © 2013 John Wiley & Sons, Ltd.     

竞争型的n元风险模型

保险市场中存在激烈的竞争,针对这种情形提出竞争型的n元风险模型,定义了两种破产时间,利用经典风险模型已有结论和条件期望的性质,得到相应的有限时间破产概率和最终破产概率表达式,以及每个保险公司有限时间破产概率和最终破产概率.