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1.
《Insurance: Mathematics and Economics》1988,7(2):75-80
In the classical compound Poisson model of the collective theory of risk let ψ(u, y) denote the probability that ruin occurs and that the negative surplus at the time of ruin is less than − y. It is shown how this function, which also measures the severity of ruin, can be calculated if the claim amount distribution is a translation of a combination of exponential distributions. Furthermore, these results can be applied to a certain discrete time model. 相似文献
2.
This paper deals with the discrete-time risk model with nonidentically distributed claims. The recursive formula of finite-time
ruin probability is obtained, which enables one to evaluate the probability of ruin with desired accuracy. Rational valued
claims and nonconstant premium payments are considered. Some numerical examples of finite-time ruin probability calculation
are presented. 相似文献
3.
We generalize an integral representation for the ruin probability in a Crámer-Lundberg risk model with shifted (or also called US-)Pareto claim sizes, obtained by Ramsay (2003), to classical Pareto(a) claim size distributions with arbitrary real values a>1 and derive its asymptotic expansion. Furthermore an integral representation for the tail of compound sums of Pareto-distributed claims is obtained and numerical illustrations of its performance in comparison to other aggregate claim approximations are provided. 相似文献
4.
In this paper,we consider a generalization of the classical ruin model,where the income is random and the distribution of the time between two claim occurrences depends on the previous claim size.This model is more appropriate than the classical ruin model.Explicit expression for the generating function of the Gerber-Shiu expected discounted penalty function are derived.A similar model is discussed.Finally,the result are showed by two examples. 相似文献
5.
Rohana S. Ambagaspitiya 《Insurance: Mathematics and Economics》2009,44(3):464-472
In this paper we relax the independence assumption of claim sizes and claim occurrence times in the Sparre Andersen model. We consider two different classes of bivariate distributions to model claim occurrence and claim sizes. We obtain explicit expressions for the ultimate ruin probability using the well known Wiener-Hopf factorization. 相似文献
6.
We consider a ruin model with random income and dependence between claim sizes and claim intervals. In this paper, we extend the determinate premium income into a compound Poisson process and assume that the distribution of the time between two claim occurrences depends on the previous claim size.Given the premium size is exponentially distributed, the(Gerber-Shiu) discounted penalty functions is derived.Finally, we consider a similar model. 相似文献
7.
Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes 总被引:3,自引:0,他引:3
The paper deals with the Sparre Andersen risk model. We study the tail behaviour of the finite-time ruin probability, Ψ(x,t), in the case of subexponential claim sizes as initial risk reserve x tends to infinity. The asymptotic formula holds uniformly for t in a corresponding region and reestablishes a formula of Tang [Tang, Q., 2004a. Asymptotics for the finite time ruin probability in the renewal model with consistent variation. Stochastic Models 20, 281–297] obtained for the class of claim distributions having consistent variation. 相似文献
8.
Hansjörg Albrecher Florin Avram 《Journal of Computational and Applied Mathematics》2010,233(10):2724-2736
In this paper we propose a highly accurate approximation procedure for ruin probabilities in the classical collective risk model, which is based on a quadrature/rational approximation procedure proposed in [2]. For a certain class of claim size distributions (which contains the completely monotone distributions) we give a theoretical justification for the method. We also show that under weaker assumptions on the claim size distribution, the method may still perform reasonably well in some cases. This in particular provides an efficient alternative to a related method proposed in [3]. A number of numerical illustrations for the performance of this procedure is provided for both completely monotone and other types of random variables. 相似文献
9.
10.
In this paper, we obtain the asymptotics for the tail probability of the total claim amount with negatively dependent claim
sizes in two cases: in the first case, the distribution tail of the claim number is dominatedly varying; in the second case,
the distribution of the claim number is in the maximum domain of attraction of the Gumbel distribution, and the claim sizes
are light-tailed. In both cases, we assume that the claim sizes are nondegenerate negatively dependent and identically distributed
random variables and that the claim number is not necessarily independent of the claim sizes. As applications, we derive asymptotics
for the finite-time ruin probabilities in some dependent compound renewal risk models with constant interest rate. 相似文献
11.
Konstantin A. Borovkov David C.M. Dickson 《Insurance: Mathematics and Economics》2008,42(3):1104-1108
We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. [Dickson, D.C.M., Hughes, B.D., Zhang, L., 2005. The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. Scand. Actuar. J., 358–376] for such processes with Erlang inter-claim times. The derivation is based on transforming the original boundary crossing problem to an equivalent one on linear lower boundary crossing by a spectrally positive Lévy process. We illustrate our result in the cases of gamma, mixed exponential and inverse Gaussian inter-claim time distributions. 相似文献
12.
In this paper, we consider the distribution of the maximum surplus before ruin in a perturbed risk model with two independent classes of risks, in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations for the distribution of the maximum surplus before ruin. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family. 相似文献
13.
In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly
study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity
is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous
integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation,
the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long
as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted
dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results. 相似文献
14.
15.
In this paper, we consider the Sparre Andersen risk model modified by the inclusion of interest on the surplus. By using the techniques of Cai and Dickson [Ins.: Math. Econ. 32(2003)], we give the functional and also the exponential type upper bounds for the tail probability of the deficit at ruin. Some special cases are also discussed. 相似文献
16.
Some reliability based properties of compound geometric distributions are derived using an approach motivated by the analysis of the deficit at ruin in a renewal risk theoretic setting. Implications for generalizing the result of Cai and Kalashnikov [J. Appl. Prob. 37 (2000) 283–289] are discussed. Subsequently, analysis of the distribution of the deficit itself in the renewal risk setting is considered. The regenerative nature of the ruin problem in the renewal risk model is exploited to study exact and approximate properties of the deficit at ruin (given that ruin occurs). Central to the discussion are the compound geometric components of the maximal aggregate loss. The proper distribution of the deficit, given that ruin occurs, is a mixture of residual ladder height distributions, from which various exact relationships and bounds follow. The asymptotic (in the initial surplus) distribution of the deficit is also considered. Stronger results are obtained with additional assumptions about the interclaim time or claim size distribution. 相似文献
17.
In the context of a Sparre Andersen risk model with arbitrary interclaim time distribution, the moments of discounted aggregate claim costs until ruin are studied. Our analysis relies on a novel generalization of the so-called discounted density which further involves a moment-based component. More specifically, while the usual discounted density contains a discount factor with respect to the time of ruin, we propose to incorporate powers of the sum until ruin of the discounted (and possibly transformed) claims into the density. Probabilistic arguments are applied to derive defective renewal equations satisfied by the moments of discounted aggregate claim costs until ruin. Detailed examples concerning the discounted aggregate claims and the number of claims until ruin are studied upon assumption on the claim severities. Numerical illustrations are also given at the end. 相似文献
18.
We consider a Markovian regime switching insurance risk model (also called Markov-modulated risk model). The closed form solutions for the joint distribution of surplus before and after ruin when the initial surplus is zero or when the claim size distributions are phase-type distributed are obtained. 相似文献
19.
David C.M. Dickson 《Insurance: Mathematics and Economics》2012,50(3):334-337
We use probabilistic arguments to derive an expression for the joint density of the time to ruin and the number of claims until ruin in the classical risk model. From this we obtain a general expression for the probability function of the number of claims until ruin. We also consider the moments of the number of claims until ruin and illustrate our results in the case of exponentially distributed individual claims. Finally, we briefly discuss joint distributions involving the surplus prior to ruin and deficit at ruin. 相似文献
20.
A. Aleškevičienė 《Lithuanian Mathematical Journal》2010,50(1):1-12
In this paper, we consider the Sparre Andersen insurance risk model with initial surplus x > 0. Let Ax denote the deficit at the time of ruin. Assuming that the tail probabilities of claim size are regularly varying, the second-order
asymptotic of the deficit distribution and its moments at the time of ruin are derived. 相似文献