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.     

A ruin model with random income and dependence between claim sizes and claim intervals

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.     

A ruin model with compound poisson income and dependence between claim sizes and claim intervals

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.     

Ultimate ruin probability in the Sparre Andersen model with dependent claim sizes and claim occurrence times

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.     

On the ruin time distribution for a Sparre Andersen process with exponential claim sizes

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.     

Recursive calculation of finite-time ruin probabilities

We develop a simple algorithm for the numerical calculation of finite-time ruin probabilities in a general discrete-time risk process model. These probabilities can be used for the calculation of approximations for the finite-time ruin probabilities in the classical actuarial risk model.     

The probability and severity of ruin for combinations of exponential claim amount distributions and their translations

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.     

Parisian ruin over a finite-time horizon

For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) 0}, S, T_u 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval.     

On ruin probability and aggregate claim representations for Pareto claim size distributions

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.     

Sensitivity analysis and density estimation for finite-time ruin probabilities

The goal of this paper is to obtain probabilistic representation formulas that are suitable for the numerical computation of the (possibly non-continuous) density functions of infima of reserve processes commonly used in insurance. In particular we show, using Monte Carlo simulations, that these representation formulas perform better than standard finite difference methods. Our approach differs from Malliavin probabilistic representation formulas which generally require more smoothness on random variables and entail the continuity of their density functions.     

Uniform asymptotics for finite-time ruin probability of a bidimensional risk model

Consider a continuous-time bidimensional risk model with constant force of interest in which the claim sizes from the same business are heavy-tailed and upper tail asymptotically independent. We investigate two cases: one is that the two claim-number processes are arbitrarily dependent, and the other is that the two corresponding claim inter-arrival times from different lines are positively quadrant dependent. Some uniformly asymptotic formulas for finite-time ruin probability are established.     

Estimates for the finite-time ruin probability with insurance and financial risks

The paper gives estimates for the finite-time ruin probability with insurance and financial risks. When the distribution of the insurance risk belongs to the class L(??) for some ?? > 0 or the subexponential distribution class, we abtain some asymptotic equivalent relationships for the finite-time ruin probability, respectively. When the distribution of the insurance risk belongs to the dominated varying-tailed distribution class, we obtain asymptotic upper bound and lower bound for the finite-time ruin probability, where for the asymptotic upper bound, we completely get rid of the restriction of mutual independence on insurance risks, and for the lower bound, we only need the insurance risks to have a weak positive association structure. The obtained results extend and improve some existing results.     

Explicit ruin formulas for models with dependence among risks

We show that a simple mixing idea allows one to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.     

Finite-time ruin probability in the inhomogeneous claim case

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.     

Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed

In the renewal risk model, several strong hypotheses may be found too restrictive to model accurately the complex evolution of the reserves of an insurance company. In the case where claim sizes are heavy-tailed, we relax the independence and stationarity assumptions and extend some asymptotic results on finite-time ruin probabilities, to take into account possible correlation crises like the one recently bred by the sub-prime crisis: claim amounts, in general assumed to be independent, may suddenly become strongly positively dependent. The impact of dependence and non-stationarity is analyzed and several concrete examples are given.     

Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed

In the renewal risk model, several strong hypotheses may be found too restrictive to model accurately the complex evolution of the reserves of an insurance company. In the case where claim sizes are heavy-tailed, we relax the independence and stationarity assumptions and extend some asymptotic results on finite-time ruin probabilities, to take into account possible correlation crises like the one recently bred by the sub-prime crisis: claim amounts, in general assumed to be independent, may suddenly become strongly positively dependent. The impact of dependence and non-stationarity is analyzed and several concrete examples are given.     

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.     

On the efficient evaluation of ruin probabilities for completely monotone claim distributions

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.     

The surpluses immediately before and at ruin,and the amount of the claim causing ruin

In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u; x, y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap(x + y), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h(u; z) can be obtained from f(u; x, y). One finds, for example, that h(0; z) = azp(z).     

Uniform estimates for the finite-time ruin probability in the dependent renewal risk model

This paper investigates the finite-time ruin probability in the dependent renewal risk model, where the claim sizes are independent and identically distributed random variables with strongly subexponential tails, and the interarrival times are negatively dependent. We establish an asymptotic estimate, which holds uniformly for the time horizon varying in the positive half line.