Asymptotics for the Moments of the Time to Ruin for the Compound Poisson Model Perturbed by Diffusion

We obtain the asymptotic behaviour of the k-th moment of the time to ruin in the classical risk model perturbed by diffusion for the case where the claim size distribution has a heavy tail.     

On average losses in the ruin problem with fractional Brownian motion as input

We consider the model , where u > 0, c > 0, is the fractional Brownian motion with Hurst parameter H, 0 < H < 1. We study the asymptotic behavior of average losses in the case of ruin, i.e. the asymptotic behavior of the conditional expected value as u→ ∞ . Three cases are considered: the short time horizon, with T finite or growing much slower than u; the long time horizon, with T at or above the time of ruin, including infinity; and the intermediate time horizon, with T proportional to u but not growing as fast as in the long time horizon. As one of the examples, we derive an asymptotically optimal portfolio minimizing average losses in the case of two independent markets. Vladimir Piterbarg was supported by RFFI grant of Russian Federation 07-01-00077.     

Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes

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.     

Ruin probability and joint distributions of some actuarial random vectors in the compound pascal model

The compound negative binomial model,introduced in this paper,is a discrete time version.We discuss the Markov properties of the surplus process,and study the ruin probability and the joint distributions of actuarial random vectors in this model.By the strong Markov property and the mass function of a defective renewal sequence,we obtain the explicit expressions of the ruin probability,the finite-horizon ruin probability,the joint distributions of T,U(T-1),|U(T)| and 0≤inn相似文献   

Asymptotic analysis of a risk process with high dividend barrier

In this paper we study a risk model with constant high dividend barrier. We apply Keilson’s (1966) results to the asymptotic distribution of the time until occurrence of a rare event in a regenerative process, and then results of the cycle maxima for random walk to obtain the asymptotic distribution of the time to ruin and the amount of dividends paid until ruin.     

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.     

Joint and supremum distributions in the compound binomial model with Markovian environment

In this paper, we study the compound binomial model in Markovian environment, which is proposed by Cossette, et al. (2003). We obtain the recursive formula of the joint distributions of T, X(T − 1) and |X(T)| (i.e., the time of ruin, the surplus before ruin and the deficit at ruin) by the method of mass function of up-crossing zero points, as given by Liu and Zhao (2007). By using the same method, the recursive formula of supremum distribution is obtained. An example is included to illustrate the results of the model.     

Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model

Consider an insurer who is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same asymptotic formula holds for the finite-time and infinite-time ruin probabilities. Restricting our attention to the so-called constant investment strategy, we show how the insurer adjusts his investment portfolio to maximize the expected terminal wealth subject to a constraint on the ruin probability.     

Asymptotics for the ruin probabilities of a two‐dimensional renewal risk model with heavy‐tailed claims

In this paper, we consider two dependent classes of insurance business with heavy‐tailed claims. The dependence comes from the assumption that claim arrivals of the two classes are governed by a common renewal counting process. We study two types of ruin in the two‐dimensional framework. For each type of ruin, we establish an asymptotic formula for the finite‐time ruin probability. These formulae possess a certain uniformity feature in the time horizon. Copyright © 2010 John Wiley & Sons, Ltd.     

On the two-scale method for the problem of perturbed one-frequency oscillations

We consider the asymptotic behavior with respect to time of the solution to the initial problem for an ordinary differential equation with a small parameter ∈. We construct an asymptotic approximation that is valid for time valuest≫∈ up to any order in ∈. Translated from Teoreticheskaya i Matematicheskay Fizika, Vol. 118, No. 3, pp. 383–389, March, 1999.     

Ruin probabilities of a bidimensional risk model with investment

We consider a classical risk model with the possibility of investment. We study two types of ruin in the bidimensional framework. Using the martingale technique, we obtain an upper bound for the infinite-time ruin probability with respect to the ruin time T_max(u₁,u₂). For each type of ruin, we derive an integral-differential equation for the survival probability, and an explicit asymptotic expression for the finite-time ruin probability.     

Ruin problems for an autoregressive risk model with dependent rates of interest

In this paper, we consider a discrete insurance risk model in which the claims, the premiums and the rates of interest are assumed to have dependent autoregressive structures (AR(1)). We derive recursive and integral equations for expected discounted penalty function. By these equations, we obtain generalized Lundberg inequality for the infinite time severity of ruin and hence for the infinite time ruin probability, consider asymptotic formula for the finite time ruin probability when loss distributions have regularly varying tails, and study some probability properties of the duration of ruin.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tail behavior of random sums under consistent variation with applications to the compound renewal risk model

In this paper, we consider the random sums of i.i.d. random variables ξ ₁,ξ ₂,... with consistent variation. Asymptotic behavior of the tail P(ξ₁ + ... + ξ_η > x), where η is independent of ξ ₁,ξ ₂,..., is obtained for different cases of the interrelationships between the tails of ξ ₁ and η. Applications to the asymptotic behavior of the finite-time ruin probability ψ(x,t) in a compound renewal risk model, earlier introduced by Tang et al. (Stat Probab Lett 52, 91–100 (2001)), are given. The asymptotic relations, as initial capital x increases, hold uniformly for t in a corresponding region. These asymptotic results are illustrated in several examples.      

常息力更新场合有限时间破产概率对负相依索赔额的不敏感性

设索赔来到过程为具有常数利息力度的更新风险模型.在索赔额分布为负相依的次指数分布假定下,建立了有限时间破产概率的一个渐近等价公式.所得结果显示,在独立同分布索赔额情形,有限时间破产概率的有关渐近等价公式,在负相依场合依然成立.这表明有限时间破产概率对于索赔额的负相依结构是不敏感的.     

Evaluation of the probability of bankruptcy for a model of insurance company

We study the problem of evaluation of the probability of ruin of an insurance company for infinitely many steps in the case where the company can invest its capital to bank deposits at any time. As a distribution of the amounts of claims to the insurance company, we use the gamma-distribution with the parameters n and α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 447–457, April, 2007.     

Bounds for classical ruin probabilities

We derive upper and lower bounds for the ruin probability over infinite time in the classical actuarial risk model (usual independence and equidistribution assumptions; the claim-number process is Poisson). Our starting point is the renewal equation for the ruin probability, but no renewal theory is used, except for the elementary facts proved in the note. Some bounds allow a very simple new proof of an asymptotic result akin to heavy-tailed claim-size distributions.     

On asymptotics of deficit distribution and its moments at the time of ruin

In this paper, we consider the Sparre Andersen insurance risk model with initial surplus x > 0. Let A_x 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.     

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.     

General tax structures for a Lévy insurance risk process under the Cramér condition

We investigate the Lévy insurance risk model with tax under Cramér’s condition. A direct analogue of Cramér’s estimate for the probability of ruin in this model is obtained, together with the asymptotic distribution, conditional on ruin occurring, of several variables of interest related to ruin including the surplus immediately prior to ruin (undershoot) and shortfall at ruin (overshoot). We also compute the present value of all tax paid conditional on ruin occurring. The proof involves first transferring results from the model with no tax to the reflected process, and from there to the model with tax.