The joint distribution of the Parisian ruin time and the number of claims until Parisian ruin in the classical risk model (original) (raw)
On the efficient evaluation of ruin probabilities for completely monotone claim distributions
Journal of Computational and Applied Mathematics, 2010
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 by Trefethen et al. [12]. 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 by Thorin [10]. A number of numerical illustrations for the performance of this procedure is provided for both completely monotone and other types of random variables.
Ruin probabilities based at claim instants for some non-Poisson claim processes
Insurance: Mathematics and Economics, 2000
The paper presents a recursive method of calculating ruin probabilities for non-Poisson claim processes, by looking at the surplus process embedded at claim instants. The developed method is exact. The processes considered have both claim sizes and the inter-claim revenue following selected phase type distributions. The numerical section contains figures derived from the exact approach, as well as a tabular example using the numerical approach of De Vylder and Goovaerts. The application of the method derived in the paper through numerical examples reveals the sensitivity of the value of probability of ruin to changes in claim number process.
Parisian Ruin With Exponential Claims
Submitted for publication, see http://stats. lse. ac. …, 2009
In this paper, we extend the concept of ruin in risk theory to the Parisian type of ruin. For this to occur, the surplus process must fall below zero and stay negative for a continuous time interval of specified length. Working with a classical surplus process with exponential ...
The density of time to ruin in Poisson risk model
We derive an expression for the density of the time to ruin in the classical risk model by inverting its Laplace transform. We then apply the result when the individual claim amount distribution is a mixed Erlang distribution, and show how finite time ruin probabilities can be calculated in this case.
Some comparison results for finite-time ruin probabilities in the classical risk model
Insurance: Mathematics and Economics, 2017
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.
2013
In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distribution -- Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs.
Approximations of the ruin probability in a discrete time risk model
arXiv: Probability, 2020
Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.
Analysis of the ruin probability using Laplace transforms and Karamata-Tauberian theorem
The classical result of Cramer-Lundberg states that if the rate of premium, c, exceeds the average of the claims paid per unit time, λµ, then the probability of ruin of an insurance company decays exponen-tially fast as the initial capital u → ∞. In this note, the asymptotic behavior of the probability of ruin is derived by means of infinitesimal generators and Laplace transforms. Using these same tools, it is shown that the probability of ruin has an algebraic decay rate if the insurance company invests its capital in a risky asset with a price which follows a geometric Brownian motion. The latter result is shown to be valid not only for exponentially distributed claim amounts, as in Frolova et al. (2002), but, more generally, for any claim amount distribution that has a moment generating function defined in a neighborhood of the origin.
Analysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems
The classical result of Cramer-Lundberg states that if the rate of premium, c, exceeds the average of the claims paid per unit time, ‚", then the probability of ruin of an insurance company decays exponen- tially fast as the initial capital u ! 1. In this note, the asymptotic behavior of the probability of ruin is derived by means of inflnitesimal generators and Laplace transforms. Using these same tools, it is shown that the probability of ruin has an algebraic decay rate if the insurance company invests its capital in a risky asset with a price which follows a geometric Brownian motion. The latter result is shown to be valid not only for exponentially distributed claim amounts, as in Frolova et al. (2002), but, more generally, for any claim amount distribution that has a moment generating function deflned in a neighborhood of the origin.
Simulating the ruin probability of risk processes with delay in claim settlement
Stochastic Processes and their Applications, 2004
A risk process with delay in claim settlement is usually described in terms of a Poisson shot-noise process (see Kl uppelberg and Mikosch (Bernoulli 1 (1995) 125) and Brà emaud (Appl. Probab. 37 (2000) 914)). In particular, Brà emaud proves that under suitable conditions the corresponding ruin probability goes to zero not slower than an exponential rate. This yields problems if we want to estimate the ruin probability by a Monte Carlo simulation. In this paper we overcome these di culties deriving the asymptotically e cient simulation law.