Showing papers in "Scandinavian Actuarial Journal in 1996"
TL;DR: In this paper, the authors consider the distributions of the duration of a single period of negative surplus and of the total duration of the negative surplus, and derive explicit results where possible and show how to approximate these distributions through the use of a discrete time risk model.
Abstract: In the classical risk model we allow the surplus process to continue if the surplus falls below zero. We consider the distributions of the duration of a single period of negative surplus and of the total duration of negative surplus. We derive explicit results where possible and show how to approximate these distributions through the use of a discrete time risk model.
44 citations
TL;DR: In this paper, a class of premium calculation principles is considered with the premiums obtained as expected values of suitably transformed distribution functions, and the likelihood ratio ordering of risks is preserved for any of these principles.
Abstract: A class of premium calculation principles is considered with the premiums obtained as expected values of suitably transformed distribution functions. The Esscher principle is a particular example. It is found that the likelihood ratio ordering of risks is preserved for any of these principles. A renewal theoretic interpretation of a special principle is given, and useful properties as well as a related characterization of the exponential distribution are shown.
43 citations
TL;DR: Finite and infinite-time classical ruin probabilities can be approximated in Gerber's elementary binomial risk model and rather rough discretizations provide approximations of excellent quality when a new claimsize distribution is adopted and when anew security loading is introduced.
Abstract: Finite and infinite-time classical ruin probabilities can be approximated in Gerber's elementary binomial risk model. In order to obtain good results, rather fine discretizations may be necessary and then the computing times may be much too long. Here we show how rather rough discretizations provide approximations of excellent quality when a new claimsize distribution (with one negative probability mass!!!) is adopted and when a new security loading is introduced.
32 citations
TL;DR: In this article, it is shown that the ruin probabilities of a risk reserve process can be expressed as the solution of a finite set of differential equations, and similar results are obtained for the case where the process evolves in a Markovian environment (e.g., a numerical example of a stochastic interest rate).
Abstract: Consider a risk reserve process with initial reserve u, Poisson arrivals, premium rule p(r) depending on the current reserve r and claim size distribution which is phase-type in the sense of Neuts. It is shown that the ruin probabilities ψ(u) can be expressed as the solution of a finite set of differential equations, and similar results are obtained for the case where the process evolves in a Markovian environment (e.g., a numerical example of a stochastic interest rate is presented). Further, an explicit formula for ψ(u) is presented for the case where p(r) is a two-step function. By duality, the results apply also to the stationary distribution of storage processes with the same input and release rate p(r) at content r.
24 citations
TL;DR: In this article, a method for obtaining two-sided bounds of ruin probabilities is proposed based on the analysis of so-called geometric sums which are sums of i.i.d.s and the number of summands is a r.v.
Abstract: A new method for obtaining two-sided bounds of ruin probabilities is proposed. This is based on the analysis of so-called geometric sums which are sums of i.i.d.r.v.'s and the number of summands is a r.v. also having a geometric distribution. The method uses probabilistic arguents only and does not use analytic relations such as Spitzer identity. It is shown that some well known bounds can be proved with the help of the developed method. Numerical examples are considered.
23 citations
TL;DR: In this paper, a simple recursive procedure for the calculation of mixed compound Poisson distributions was proposed, where the logarithm of the mixing density can be written as the ratio of two polynomials.
Abstract: We derive a simple recursive procedure for calculation of mixed compound Poisson distributions, when the logarithm of the mixing density can be written as the ratio of two polynomials.
16 citations
TL;DR: In this article, the classical Thiele's differential equation for the prospective reserve of an insurance policy has been generalized to models with counting process driven payments and deterministic interest, and the technique of proof consists in identifying the null part of the martingale associated with the initial present value of the payments.
Abstract: The classical Thiele's differential equation for the prospective reserve of an insurance policy has been generalized to models with counting process driven payments and deterministic interest. Here the result is extended to situations with diffusion driven stochastic interest. The technique of proof consists in identifying the null part of the martingale associated with the initial present value of the payments. The presentation centers on life insurance, but the theory can be adapted to more general stochastic payment streams.
15 citations
TL;DR: Theorem 3 of Section 3 of the paper as mentioned in this paper requires that the statewise reserves Vj be precisely defined, and it turns out that Theorem 3 as stated may require that an appropriate definition be used.
Abstract: A. Purpose of this note Any specific application of the theory in Section 3 of the paper would demand that the statewise reserves Vj be precisely defined. There is some latitude at this point, however, and it turns out that Theorem 3 as stated may require that an appropriate definition be used. Paragraph B of the present note adds rigour on the issue. Paragraph C offers some guidance as to how to construct and compute the reserves in nontrivial cases. Some technical lemmas are placed in the final Paragraph D.
11 citations
TL;DR: In this article, a new asymptotic expression and upper bounds on probabilities of ruin after time t(u) ≫ and before time 0 < t(U) ≪, as the initial risk reserve u increases to infinity, are suggested.
Abstract: In the framework of Andersen's risk model, a new asymptotic expression and upper bounds on probabilities of ruin after time t(u) ≫ and before time 0 < t(u) ≪ , as the initial risk reserve u increases to infinity, are suggested. This result complements the classical normal-type approximation for the probability of ruin within finite time and is designed as its large deviations counterpart. The main technical device of the paper (see Section 3), which is of independent interest, are the upper bounds and the asymptotic expressions for the probabilities of large deviations of the stopped random walks, developed under low moment conditions.
9 citations
TL;DR: In this article, the transient behavior of a stochastic model for high demand continuing care retirement community (CCRC) populations is examined, where a time-homogeneous Markov process is used to model the care requirements of the CCRC residents.
Abstract: This paper examines the transient behavior of a stochastic model for high demand continuing care retirement community (CCRC) populations. The CCRC under consideration provides a number of independent living units each of which houses one resident at a time and a skilled nursing facility (SNF) for those who require care. A time-homogeneous Markov process is used to model the care requirements of the CCRC residents. Under the model, the “state” of the population is described by the number of permanent transfers and the number of temporary transfers to the SNF. The paper examines how one can obtain information about the distribution of the state of the CCRC population at a given future time.
8 citations
TL;DR: Asymptotic normality of nonparametric estimators is derived using the delta-method and Pollard's Central Limit Theorem for the empirical process indexed by a class of functions as discussed by the authors.
Abstract: Asymptotic normality of nonparametric estimators is derived using the delta-method and Pollard's Central Limit Theorem for the empirical process indexed by a class of functions. The results are applied to estimation problems in actuarial mathematics.
TL;DR: In this article, an extension of this kind of models, by investigating the situation of interest rates that cannot become negative, is presented, where the case of an annuity certain and in particular that of a perpetuity is dealt with in detail.
Abstract: Recently, the authors showed how interest randomness in actuarial functions can· be described by means of Wiener processes using path integrals. This paper wants to present an extension of this kind of models, by investigating the situation of interest rates that cannot become negative. The case of an annuity certain and in particular that of a perpetuity will be dealt with in detail.
TL;DR: In this article, the authors correct an error in the second step of their treatment of the compound generalized Poisson and negative binomial distributions, and give a three-step recursive algorithm for the evaluation of terms in compound generalized distributions.
Abstract: Kling & Goovaerts ( 1993) have given a three-step recursive algorithm for the evaluation of terms in compound generalized distributions. This note corrects an error in the second step of their treatment of the compound generalized Poisson and negative binomial distributions.
TL;DR: In this article, the problem of constructing confidence interval for the largest k normal mean was extended by considering the case when the populations have unequal and unknown variances and several multi-stage estimation procedures (like, two-stage, three-stage and accelerated sequential procedures) were developed to deal with the estimation problem.
Abstract: The problem of constructing confidence interval for the largest of k normal means considered by Saxena and Tong (1969) and Tong (1970, 1973) is extended by considering the case when the populations have unequal and unknown variances. Several multi-stage estimation procedures (like, two-stage, three-stage and accelerated sequential procedures) are developed to deal with the estimation problem. Nice asymptotic properties of these estimation procedures are studied and their relative advantages and disadvantages are discussed.
TL;DR: The stochastic approach to road transportation of dangerous materials as well as pollution is described, based on the same ideas as used in ‘A stochastically approach to insurance cycles’ by Goovaerts et al. ( 1992).
Abstract: Road transportation of dangerous materials as well as pollution are considered as being catastrophic risks. In case the normal insurance models are applied to catastrophic risks, one of the basic assumptions, namely that the average surplus is a linear quantity in time, is certainly not satisfied. In more realistic situations, an average movement of the surplus is linear up to a certain time, where a drastic decrease in the surplus occurs, namely at the moment when the claims are paid out, then afterwards the surplus will show a linear trend again. In the present contribution we describe the stochastic approach to this kind of situations, based on the same ideas as used in ‘A stochastic approach to insurance cycles’ by Goovaerts et al. ( 1992). The consequences for the probability of ruin at a specific point in time are also investigated as well as their relation to solvency margins.