Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications

The aggregate claim amount in a particular time period is a quantity of fundamental importance for proper management of an insurance company and also for pricing of insurance coverages. In this paper, we show that the proportional hazard rates (PHR) model, which includes some well-known distributions such as exponential, Weibull and Pareto distributions, can be used as the aggregate claim amount distribution. We also present some conditions for the use of exponentiated Weibull distribution as the claim amount distribution. The results established here complete and extend the well-known result of Khaledi and Ahmadi (2008).     

On the analysis of time dependent claims in a class of birth process claim count models

An integral representation is derived for the sum of all claims over a finite interval when the claim value depends upon its incurral time. These time dependent claims, which generalize the usual compound model for aggregate claims, have insurance applications involving models for inflation and payment delays. The number of claims process is assumed to be a (possibly delayed) nonhomogeneous birth process, which includes the Poisson process, contagion models, and the mixed Poisson process, as special cases. Known simplified compound representations in these special cases are easily generalized to the conditional case, given the number of claims at the beginning of the interval. Applications to the case involving “two stages” are also considered.     

Applying copula models to individual claim loss reserving methods

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Recently, individual claim loss models have attracted a great deal of interest in the actuarial literature, which overcome some shortcomings of aggregated claim loss models. The dependence of the event times with the delays is a crucial issue for estimating the claim loss reserving. In this article, we propose to use semi-competing risks copula and semi-survival copula models to fit the dependence structure of the event times with delays in the individual claim loss model. A nonstandard two-step procedure is applied to our setting in which the associate parameter and one margin are estimated based on an ad hoc estimator of the other margin. The asymptotic properties of the estimators are established as well. A simulation study is carried out to evaluate the performance of the proposed methods.     

Mathematical investigation of the Gerber-Shiu function in the case of dependent inter-claim time and claim size

In this paper we investigate the well-known Gerber-Shiu expected discounted penalty function in the case of dependence between the inter-claim times and the claim amounts. We set up an integral equation for it and we prove the existence and uniqueness of its solution in the set of bounded functions. We show that if δ>0, the limit property of the solution is not a regularity condition, but the characteristic of the solution even in the case when the net profit condition is not fulfilled. It is the consequence of the choice of the penalty function for a given density function. We present an example when the Gerber-Shiu function is not bounded, consequently, it does not tend to zero. Using an operator technique we also prove exponential boundedness.     

On a risk model with claim investigation

In this paper, a queue-based claims investigation mechanism is considered to model an insurer’s claim processing practices. The resulting risk model may be viewed as a first step in developing models with more realistic claim investigation mechanisms. Related to claim investigations, claim settlement delays and time dependent payments have been studied in a ruin context by, e.g. Taylor (1979), Cai and Dickson (2002), and Trufin et al. (2011). However, little has been done on queue-based investigation mechanisms. We first demonstrate the impact of a particular claim investigation system on some common ruin-related quantities when claims arrive according to a compound Poisson process, and investigation times are of a combination of exponential form. Probabilistic interpretations for the defective renewal equation components are also provided. Finally, via numerical examples, we explore various risk management questions related to this problem such as how claim investigation strategies can help an insurer control its activities within its risk appetite.     

Surplus analysis for a class of Coxian interclaim time distributions with applications to mixed Erlang claim amounts

Gerber-Shiu analysis with the generalized penalty function proposed by Cheung et al. (in press-a) is considered in the Sparre Andersen risk model with a K_n family distribution for the interclaim time. A defective renewal equation and its solution for the present Gerber-Shiu function are derived, and their forms are natural for analysis which jointly involves the time of ruin and the surplus immediately prior to ruin. The results are then used to find explicit expressions for various defective joint and marginal densities, including those involving the claim causing ruin and the last interclaim time before ruin. The case with mixed Erlang claim amounts is considered in some detail.     

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).     

Explicit Expressions for the Ruin Probabilities of Erlang Risk Processes with Pareto Individual Claim Distributions

In this paper we first consider a risk process in which claim inter-arrival times and the time untilthe first claim have an Erlang (2) distribution.An explicit solution is derived for the probability of ultimateruin,given an initial reserve of u when the claim size follows a Pareto distribution.Follow Ramsay,Laplacetransforms and exponential integrals are used to derive the solution,which involves a single integral of realvalued functions along the positive real line,and the integrand is not of an oscillating kind.Then we showthat the ultimate ruin probability can be expressed as the sum of expected values of functions of two differentGamma random variables.Finally,the results are extended to the Erlang(n) case.Numerical examples aregiven to illustrate the main results.     

Knapp-Wallach Szegö integrals and generalized principal series representations: The parabolic rank one case

To each discrete series representation of a connected semisimple Lie group G with finite center, a G-equivariant embedding into a generalized principal series representation is given. This representation is induced from specified parameters on a maximal parabolic subgroup of G and the mapping is defined by an integral formula, analogous to the Szegö integral introduced by Knapp and Wallach for a minimal parabolic subgroup. In a limiting case, embeddings of limits of discrete series representations are obtained and used to exhibit a reducibility theorem.     

Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts

The asymptotic normality of the sample proportional hazard premium for heavy-tailed claim amounts with infinite variance cannot be obtained by classical results for L-statistics. In this paper, we propose an alternative estimator for this class of premiums and we establish its asymptotic normality.     

On a risk model with random incomes and dependence between claim sizes and claim intervals

In this paper, we construct a risk model with a dependence setting where there exists a specific structure among the time between two claim occurrences, premium sizes and claim sizes. Given that the premium size is exponentially distributed, both the Laplace transforms and defective renewal equations for the expected discounted penalty functions are obtained. Exact representations for the solutions of the defective renewal equations are derived through an associated compound geometric distribution. When the claims are subexponentially distributed, the asymptotic formulae for ruin probabilities are obtained. Finally, when the individual premium sizes have rational Laplace transforms, the Laplace transforms for the expected discounted penalty functions are obtained.     

Risk model with fuzzy random individual claim amount

In this paper, we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a Poisson process. The mean chance of the ultimate ruin is researched. Particularly, the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable. The results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables. Finally, two numerical examples are presented.     

Atomic maps and the Chogoshvili-Pontrjagin claim

It is proved that all spaces of dimension three or more disobey the Chogoshvili-Pontrjagin claim. This is of particular interest in view of the recent proof (in Certain 2-stable embeddings, by Dobrowolski, Levin, and Rubin, Topology Appl. 80 (1997), 81-90) that two-dimensional ANRs obey the claim.

The construction utilizes the properties of atomic maps which are maps whose fibers (point inverses) are atoms (hereditarily indecomposable continua).

A construction of M. Brown is applied to prove that every finite dimensional compact space admits an atomic map with a one-dimensional range.

     

A process with stochastic claim frequency and a linear dividend barrier

The classical model of ruin theory is given by a Poisson claim number process with single claims X_i and constant premium flow. Gerber has generalized this model by a linear dividend barrier b+at. Whenever the free reserve of the insurance reaches the barrier, dividends are paid out in such a way that the reserve stays on the barrier. The aim of this paper is to give a generalization of this model by using the idea of Reinhard. After an exponentially distributed time, the claim frequency changes to a different level, and can change back again in the same way. This may be used e.g. in storm damage insurance. The computations lead to systems of partial integro differential equations which are solved.     

A vector measure approach to state feedback control in Banach resolution spaces I

An abstract version of the linear regulator-quadratic cost problem is considered for a dynamical system S, where input and output are elements of various Banach resolution spaces. Our main result is the representation of the optimal control in memoryless state feedback form. This representation is obtained as an integral with respect to a vector measure defined on the state space of S.     

TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts

In this paper, we consider a portfolio of n dependent risks X₁,…,X_n and we study the stochastic behavior of the aggregate claim amount S=X₁+?+X_n. Our objective is to determine the amount of economic capital needed for the whole portfolio and to compute the amount of capital to be allocated to each risk X₁,…,X_n. To do so, we use a top-down approach. For (X₁,…,X_n), we consider risk models based on multivariate compound distributions defined with a multivariate counting distribution. We use the TVaR to evaluate the total capital requirement of the portfolio based on the distribution of S, and we use the TVaR-based capital allocation method to quantify the contribution of each risk. To simplify the presentation, the claim amounts are assumed to be continuously distributed. For multivariate compound distributions with continuous claim amounts, we provide general formulas for the cumulative distribution function of S, for the TVaR of S and the contribution to each risk. We obtain closed-form expressions for those quantities for multivariate compound distributions with gamma and mixed Erlang claim amounts. Finally, we treat in detail the multivariate compound Poisson distribution case. Numerical examples are provided in order to examine the impact of the dependence relation on the TVaR of S, the contribution to each risk of the portfolio, and the benefit of the aggregation of several risks.     

Functional integral representations of the Pauli-Fierz model with spin 1/2

A Feynman-Kac-type formula for a Lévy and an infinite-dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e−tHPF generated by the Pauli-Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics is constructed. When no external potential is applied HPF turns translation-invariant and it is decomposed as a direct integral . The functional integral representation of e−tHPF(P) is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived.     

Markov dilation of Diffusion Type Processes and Its Application to the Financial Mathematics

The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for the obtained Markov processes are described. Some applications to the integral representation for functionals of diffusion type processes and to the construction of a replicating portfolio for a non-terminal contingent claim are considered.     

On a simple quasi-Monte Carlo approach for classical ultimate ruin probabilities

This note discusses a simple quasi-Monte Carlo method to evaluate numerically the ultimate ruin probability in the classical compound Poisson risk model. The key point is the Pollaczek–Khintchine representation of the non-ruin probability as a series of convolutions. Our suggestion is to truncate the series at some appropriate level and to evaluate the remaining convolution integrals by quasi-Monte Carlo techniques. For illustration, this approximation procedure is applied when claim sizes have an exponential or generalized Pareto distribution.     

Linear transmission problem for a Bitsadze system with a supersingular circle

For a second-order elliptic system with a supersingular circle whose leading part is the Bitsadze operator, we obtain an integral representation of the solution and the corresponding inversion formulas in the finite and infinite domains. The obtained integral representations are used to study the behavior of solutions as r → R and analyze a problem of linear transmission type.