Approximations of the ruin probability in a discrete time risk model (original) (raw)
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Thailand Statistician, 2017
In this paper we have discussed the infinite time ruin probabilities in continuous time in a compound Poisson process with a constant premium rate for the mixture of exponential claims. Firstly, we have fitted the mixture of two exponential and the mixture of three exponential to a set of claim data and thereafter, have computed the probability of ultimate ruin through a method giving its exact expression and then through a numerical method, namely the method of product integration. The derivation of the exact expression for ultimate ruin probability for the mixture of three and mixture of two exponential is done through the moment generating function of the maximal aggregate loss random variable. Consistencies are observed in the values of ultimate ruin probabilities obtained by both the methods.
The Probability of Eventual Ruin in the Compound Binomial Model
ASTIN Bulletin, 1989
This paper derives several formulas for the probability of eventual ruin in a discrete-time model. In this model, the number of claims process is assumed to be binomial. The claim amounts, premium rate and initial surplus are assumed to be integer-valued.
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Modern Stochastics: Theory and Applications, 2015
We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.
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.
Ruin probabilities based at claim instants for some non-Poisson claim processes
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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.
Nonparametric Estimation of the Ruin Probability in the Classical Compound Poisson Risk Model
2020
In this paper we study estimating ruin probability which is an important problem in insurance. Our work is developed upon the existing nonparametric estimation method for the ruin probability in the classical risk model, which employs the Fourier transform but requires smoothing on the density of the sizes of claims. We propose a nonparametric estimation approach which does not involve smoothing and thus is free of the bandwidth choice. Compared with the Fourier-transformation-based estimators, our estimators have simpler forms and thus are easier to calculate. We establish asymptotic distributions of our estimators, which allows us to consistently estimate the asymptotic variances of our estimators with the plug-in principle and enables interval estimates of the ruin probability.
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.
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.
Some results of ruin probability for the classical risk process
Journal of Applied Mathematics and Decision Sciences, 2003
The computation of ruin probability is an important problem in the collective risk theory. It has applications in the fields of insurance, actuarial science, and economics. Many mathematical models have been introduced to simulate business activities and ruin probability is studied based on these models. Two of these models are the classical risk model and the Cox model. In the classical model, the counting process is a Poisson process and in the Cox model, the counting process is a Cox process. Thorin (1973) studied the ruin probability based on the classical model with the assumption that random sequence followed theΓdistribution with density functionf(x)=x1β−1β1βΓ(1/β)e−xβ,x>0, whereβ>1. This paper studies the ruin probability of the classical model where the random sequence follows theΓdistribution with density functionf(x)=αnΓ(n)xn−1e−αx,x>0, whereα>0andn≥2is a positive integer. An intermediate general result is given and a complete solution is provided forn=2. Simu...