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).     

Ruin problems with compounding assets

We consider a generalization of the classical model of collective risk theory. It is assumed that the cumulative income of a firm is given by a process X with stationary independent increments, and that interest is earned continuously on the firm's assets. Then Y(t), the assets of the firm at time t, can be represented by a simple path-wise integral with respect to the income process X. A general characterization is obtained for the probability r(y) that assets will ever fall to zero when the initial asset level is y (the probability of ruin). From this we obtain a general upper bound for r(y), a general solution for the case where X has no negative jumps, and explicit formulas for three particular examples.In addition, an approximation theorem is proved using the weak convergence theory for stochastic processes. This shows that if the income process is well approximated by Brownian motion with drift, then the assets process Y is well approximated by a certain diffusion process Y₁, and r(y) is well approximated by a corresponding first passage probability r₁(y). The diffusion Y₁, which we call compounding Brownian motion, is closely related to the classical Ornstein-Uhlenbeck process.     

Parametric scaling of mathematical models

Parametric scaling, the process of extrapolation of a modelling result to new parametric conditions, is often required in model optimization, and can be important if the effects of parametric uncertainty on model predictions are to be quantified. Knowledge of the functional relationship between the model solution (y) and the system parameters (α) may also provide insight into the physical system underlying the model. This paper examines strategies for parametric scaling, assuming that only the nominal model solution y(α) and the associated parametric sensitivity coefficients (?y/?α, ?²y/?α², etc.) are known. The truncated Taylor series is shown to be a poor choice for parametric scaling, when y has known bounds. Alternate formulae are proposed which ‘build-in’ the constraints on y, thus expanding the parametric region in which the extrapolation may be valid. In the case where y has a temporal as well as a parametric dependence, the extrapolation may be further improved by removing from the Taylor series coefficients the ‘secular’ components, which refer to changes in the time scale of y(t), not to changes in y as a function of α.     

Ruin probabilities of a bidimensional risk model with investment

We consider a classical risk model with the possibility of investment. We study two types of ruin in the bidimensional framework. Using the martingale technique, we obtain an upper bound for the infinite-time ruin probability with respect to the ruin time T_max(u₁,u₂). For each type of ruin, we derive an integral-differential equation for the survival probability, and an explicit asymptotic expression for the finite-time ruin probability.     

Ruin probabilities in the discrete time renewal risk model

In this paper, we study the discrete time renewal risk model, an extension to Gerber’s compound binomial model. Under the framework of this extension, we study the aggregate claim amount process and both finite-time and infinite-time ruin probabilities. For completeness, we derive an upper bound and an asymptotic expression for the infinite-time ruin probabilities in this risk model. Also, we demonstrate that the proposed extension can be used to approximate the continuous time renewal risk model (also known as the Sparre Andersen risk model) as Gerber’s compound binomial model has been proposed as a discrete-time version of the classical compound Poisson risk model. This allows us to derive both numerical upper and lower bounds for the infinite-time ruin probabilities defined in the continuous time risk model from their equivalents under the discrete time renewal risk model. Finally, the numerical algorithm proposed to compute infinite-time ruin probabilities in the discrete time renewal risk model is also applied in some of its extensions.     

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 submartingale assumption in risk theory

The main consequences of the submartingale assumption are examined in a discrete time model. After we have shown how the submartingale decomposition theorem can be given an actuarial meaning, we formulate the ruin theory by classifying the gain processes according to the properties of the set of the safety indexes of their increments. Inequalities for ruin probabilities are derived for embeddable submartingales and for P-submartingales. As an example, we describe a rather general risk model with an adjustable gain process and we show how it can be modified to obtain examples of embeddable submartingales and P-submartingales. Optional gain processes, in which the number of policies in the portfolio is depending on the charged premiums, are also shown to satisfy the submartingale assumption; suitable demand functions can be introduced. We shortly discuss the construction of decision models to define the pricing policy of the insurance company and we provide a simple example of a pricing model with stochastic demand function.     

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.     

Ruin probability and time of ruin with a proportional reinsurance threshold strategy

In this paper, we present a threshold proportional reinsurance strategy and we analyze the effect on some solvency measures: ruin probability and time of ruin. This dynamic reinsurance strategy assumes a retention level that is not constant and depends on the level of the surplus. In a model with inter-occurrence times being generalized Erlang(n)-distributed, we obtain the integro-differential equation for the Gerber?CShiu function. Then, we present the solution for inter-occurrence times exponentially distributed and claim amount phase-type(N). Some examples for exponential and phase-type(2) claim amount are presented. Finally, we show some comparisons between threshold reinsurance and proportional reinsurance.     

Semigroup identities and proofs

Tamura proved that for any semigroup word U(x, y), if every group satisfying an identity of the form yx ~ xU(x, y)y is abelian, then so is every semigroup that satisfies that identity. Because a group has an identity element and the cancellation property, it is easier to show that a group is abelian than that a semigroup is. If we know that it is, then there must be a sequence of substitutions using xU(x, y)y ~ yx that transforms xy to yx. We examine such sequences and propose finding them as a challenge to proof by computer. Also, every model of y ~ xU(x, y)x is a group. This raises a similar challenge, which we explore in the special case y ~ x ^m y ^p x ⁿ . In addition, we determine the free model with two generators of some of these identities. In particular, we find that the free model for y ~ x ² yx ² has order 32 and is the product of D ₄ (the symmetries of a square), C ₂, and C ₄, and point out relations between such identities and Burnside’s Problem concerning models of x ⁿ ~ y ⁿ . We also examine several identities not related to groups.     

The compound binomial model with randomly paying dividends to shareholders and policyholders

Considering surplus of a joint stock insurance company based on compound binomial model, set up thresholds a₁, a₂ for shareholders and policyholders respectively. When surplus is no less than the thresholds, the company randomly pays dividends to shareholders and policyholders with probabilities q₁, q₂ respectively. For this model, we have derived the recursive formulas of both the expected discount penalty function and ruin probability, and the distribution function of the deficit at ruin.     

The Gerber-Shiu function and the generalized Cramér-Lundberg model

In this paper we extend some results in Cramér [7] by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these amounts, they are interpreted either as claims made by insureds or as income from deceased annuitants, respectively. We then demonstrate that when the events’ arrival process is a renewal process, the Gerber-Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and annuity-related income has Erlang(n, β) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation.     

Ruin probability and joint distributions of some actuarial random vectors in the compound pascal model

The compound negative binomial model,introduced in this paper,is a discrete time version.We discuss the Markov properties of the surplus process,and study the ruin probability and the joint distributions of actuarial random vectors in this model.By the strong Markov property and the mass function of a defective renewal sequence,we obtain the explicit expressions of the ruin probability,the finite-horizon ruin probability,the joint distributions of T,U(T-1),|U(T)| and 0≤inn相似文献   

Efficient simulation of finite horizon problems in queueing and insurance risk

Let ψ(u,t) be the probability that the workload in an initially empty M/G/1 queue exceeds u at time t<∞, or, equivalently, the ruin probability in the classical Crámer-Lundberg model. Assuming service times/claim sizes to be subexponential, various Monte Carlo estimators for ψ(u,t) are suggested. A key idea behind the estimators is conditional Monte Carlo. Variance estimates are derived in the regularly varying case, the efficiencies are compared numerically and also the estimators are shown to have bounded relative error in some main cases. In part, also extensions to general Lévy processes are treated.     

Optimal dividend payout for classical risk model with risk constraint

In this paper we consider the problem of maximizing the total discounted utility of dividend payments for a Cramér-Lundberg risk model subject to both proportional and fixed transaction costs.We assume that dividend payments are prohibited unless the surplus of insurance company has reached a level b.Given fixed level b,we derive a integro-differential equation satisfied by the value function.By solving this equation we obtain the analytical solutions of the value function and the optimal dividend strategy when claims are exponentially distributed.Finally we show how the threshold b can be determined so that the expected ruin time is not less than some T.Also,numerical examples are presented to illustrate our results.     

The x-and-y-axes travelling salesman problem

The x-and-y-axes travelling salesman problem forms a special case of the Euclidean TSP, where all cities are situated on the x-axis and on the y-axis of an orthogonal coordinate system of the Euclidean plane. By carefully analyzing the underlying combinatorial and geometric structures, we show that this problem can be solved in polynomial time. The running time of the resulting algorithm is quadratic in the number of cities.     

Integrability of a SIS model

We prove that the classical model of an infectious disseise, which never kills and which does not induce autoimmunity, is integrable. This model can be written as x^′=−bxy−mx+cy+mk, y^′=bxy−(m+c)y with parameters b,c,k,m∈R. We provide the explicit expression of its first integrals and of the set of all its invariant algebraic curves.     

A Diffusion Perturbed Risk Process with Stochastic Return on Investments

Abstract

A general risk model that allows for stochastic return on investments as well as perturbation by diffusion is studied. Integro-differential equations for the distributions of the time of ruin, the surplus prior to ruin and the deficit at ruin of this model are established. In particular, we consider a diffusion perturbed risk model with interest force in details.     

Two finite embedding theorems for partial 3-quasigroups

In this paper it is shown that a finite partial (x, x, y) = y 3-quasigroup can be embedded in a finite (x, x, y) = y 3-quasigroup. This result coupled with the technique of proof is then used to show that a finite partial almost Steiner 3-quasigroup {(x, x, y) = y, (x, y, z) = (x, z, y) = (y, x, z)} can be embedded in a finite almost Steiner 3-quasigroup. Almost Steiner 3-quasigroups are of more than passing interest because just like Steiner 3-quasigroups ( = Steiner quadruple systems) all of their derived quasigroups are Steiner quasigroups.     

Dividends in finite time horizon

In this paper, we consider the classical risk model modified in two different ways by the inclusion of a dividend barrier. For Model I, we present numerical algorithms, which can be used to approximate or bound the expected discounted value of dividends up to a finite time horizon, t, or ruin if this occurs earlier. We extend this by requiring the shareholders to provide the initial capital and to pay the deficit at ruin each time it occurs so that the process then continues after ruin up to time t. For Model I, we assume the full premium income is paid as dividends whenever the surplus exceeds a set level. In our Model II, we assume dividends are paid at a rate less than the rate of premium income. Copyright © 2012 John Wiley & Sons, Ltd.