Optimal dividends with incomplete information in the dual model

In [Gerber, H.U., Shiu, E.S.W., Smith, N., 2008. Methods for estimating the optimal dividend barrier and the probability of ruin. Insurance: Math. Econ. 42 (1), 243-254], methods were analyzed for estimating the optimal dividend barrier (in the sense of de Finetti). In particular, De Vylder approximations and diffusion approximations are discussed. These methods are useful when only the first few moments of the claim amount distribution are known.The purpose of this paper is to examine these and other methods (such as the gamma approximations and the gamproc approximations) in the dual model, see [Avanzi, B., Gerber, H.U., Shiu, E.S., 2007. Optimal dividends in the dual model. Insurance: Math. Econ. 41 (1), 111-123]. The dual model is obtained if the roles of premiums and claims are exchanged. In other words, the company has random gains, which constitute a compound Poisson process, and expenses occur continuously at a constant rate. The approximations can easily be implemented, and their accuracy is surprisingly good. Several numerical illustrations enhance the paper.     

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by [De Vylder and Goovaerts, 1999] and [De Vylder and Goovaerts, 2000]. First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.     

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by De Vylder and Goovaerts, 1999, De Vylder and Goovaerts, 2000. First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.     

更新模型中的Schmitter问题

Schmitter问题是指在期望和方差固定的条件下，求出使得破产概率最大（小）的索赔分布，F．De Vylder讨论了古典风险模型的情况，本文讨论了更新风险模型的情况，推广了其结果。     

Towards better multi-class parametric-decomposition approximations for open queueing networks

Methods are developed for approximately characterizing the departure process of each customer class from a multi-class single-server queue with unlimited waiting space and the first-in-first-out service discipline. The model is (GT _i /GI _i )/1 with a non-Poisson renewal arrival process and a non-exponential service-time distribution for each class. The methods provide a basis for improving parametric-decomposition approximations for analyzing non-Markov open queueing networks with multiple classes. For example, parametric-decomposition approximations are used in the Queueing Network Analyzer (QNA). The specific approximations here extend ones developed by Bitran and Tirupati [5]. For example, the effect of class-dependent service times is considered here. With all procedures proposed here, the approximate variability parameter of the departure process of each class is a linear function of the variability parameters of the arrival processes of all the classes served at that queue, thus ensuring that the final arrival variability parameters in a general open network can be calculated by solving a system of linear equations.     

Refining diffusion approximations for queues

This note illustrates the need to refine diffusion approximations for queues. Diffusion approximations are developed in several different ways for the mean waiting time in a GI/G/1 queue, yielding different results, all of which fail obvious consistency checks with bounds and exact values.     

Comparison of ruin probability estimates in the presence of heavy tails

Four methods of estimation of the ruin probability in the presence of heavy tails are compared in accordance with their effectiveness on the Pareto-distributed claim sizes. The first method, proposed by Embrechis and Veraverbeke, provides an asymptotic expression when the initial capital tends to infinity. The second method, proposed by Goovaerts and De Vylder, provides an algorithm for two-sided estimation based on the solution of the renewal equation through discretization. Its advantage is the perfect applicbility in cases with small initial capital. The third method, proposed initially by Willmoi and improved by Kalashnikov, provides an upper bound of the ruin probability with the help of a test function. The last method, proposed by Kalashnikov, provides two-sided bounds for the ruin probability using truncation of random variables to avoid the Cramér condition. The last two methods enable one to handle cases with intermediate initial capital. Proceedings of the Seminar on Stability Problems for stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Saddlepoint approximations to P-values for comparison of density estimates

Summary  This article is concerned with computing approximate p-values for the maximum of the absolute difference between kernel density estimates. The approximations are based on treating the process of local extrema of the differences as a nonhomogeneous Poisson Process and estimating the corresponding local intensity function. The process of local extrema is characterized by the intensity function, which determines the rate of local extrema above a given threshold. A key idea of this article is to provide methods for more accurate estimation of the intensity function by using saddlepoint approximations for the joint density of the difference between kernel density estimates and using the first and second derivative of the difference. In this article, saddlepoint approximations are compared to gaussian approximations. Simulation results from saddlepoint approximations show consistently better agreement between empirical p-value and predetermined value with various bandwidths of kernel density estimates.     

Approximations for the Long-Term Behavior of an Open-Population Epidemic Model

A simple stochastic epidemic model incorporating births into the susceptible class is considered. An approximation is derived for the mean duration of the epidemic. It is proved that the epidemic ultimately dies out with probability 1. The limiting behavior of the epidemic conditional on non-extinction is studied using approximation methods. Two different diffusion approximations are described and compared.     

线性过程的强逼近

该文对由独立同分布随机变量序列所生成的线性过程建立了泛函重对数律和用Wiener过程对线性过程的强逼近结果.     

Representation theorems for extremal distributions

We give a new proof of a result of Taylor and De Vylder on the representation of extremal distributions and we show how this result can be generalized.     

Moment Solutions for the State Exiting Counting Processes of a Markov Renewal Process

Important performance measures for many Markov renewal processes are the counts of the exits from each state. We present solutions for the conditional first, second, and covariance moments of the state exiting counting processes for a Markov renewal process, and solutions for the unconditional equilibrium versions of the moments. We demonstrate the relationship between the conditional first moments for the state exiting and the state entering counting processes. For analytical and illustrative purposes, we concentrate on the two state case. Two asymptotic expansions for the moment functions are proposed and evaluated both analytically and empirically. The two approximations are shown to be competitive in terms of absolute relative error, but the second approximation has a simpler analytical form which is useful in analyzing more complex stochastic processes having an underlying MRP structure.     

Risk process approximation with mixing

The approximations of risk processes with mixed exponentially distributed inter-arrival times are investigated. The number of claims in a fixed time interval is Mixed Poisson distributed. The approximating process is always overdispersed. This allows a better fit to more realistic situations in finances, than e.g. classical Cramér–Lundberg model.The claim sizes are divided in three different groups, dependently on finiteness of their first two moments. We illustrate all the cases by numerical examples.The case of diffusion approximation is investigated. Both American and European Pareto claims sizes are also studied.     

Chernoff approximations of Feller semigroups in Riemannian manifolds

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.     

Linear Programming Approximations for Markov Control Processes in Metric Spaces

We develop a general framework to analyze the convergence of linear-programming approximations for Markov control processes in metric spaces. The approximations are based on aggregation and relaxation of constraints, as well as inner approximations of the decision variables. In particular, conditions are given under which the control problems optimal value can be approximated by a sequence of finite-dimensional linear programs.     

New asymptotic representations of the noncentral t-distribution

New asymptotic approximations of the noncentral t distribution are given a generalization of the Student's t distribution. Using new integral representations, we give new asymptotic expansions not only for large values of the noncentrality parameter but also for large values of the degrees of freedom parameter. In some cases, we accept more than one large parameter. These results are not only in terms of elementary functions, but also in terms of the complementary error function and the incomplete gamma function. A number of numerical tests demonstrate the performance of the asymptotic approximations.     

Error estimates for approximate approximations with Gaussian kernels on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact intervals, only. This method, which is based on an approximate partition of unity, was introduced by Maz’ya in 1991 and has mainly been used for functions defined on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L₁-estimates are given, where all the constants are determined explicitly.     

二维风险模型中Copula 相依下的破产概率

在本文中, 我们把Copula 连结函数用到二维的风险模型中, 考虑两个模型索赔额之间基于Copula 的相依关系. 首先对二维复合Poisson 模型给出了最早破产时刻定义下的生存概率满足的偏微分方程; 然后对二维的复合二项模型, 分别在连续型索赔额分布和离散型索赔额分布下给出了不同定义的生存概率和破产概率的递归公式, 并且特别选择了FGM Copula 连结函数, 给出了相应的结果; 另外在离散型分布下, 对于其Copula 函数的不唯一性进行了说明.     

Bounds and approximations for continuous-time Markovian transition probabilities and large systems

We propose new bounds and approximations for the transition probabilities of a continuous-time Markov process with finite but large state-space. The bounding and approximating procedures have been exposed in another paper [S. Mercier, Numerical bounds for semi-Markovian quantities and applications to reliability, in revision for Methodology and Computing in Applied Probability] in the more general context of a continuous-time semi-Markov process with countable state-space. Such procedures are here specialized to the Markovian finite case, leading to much simpler algorithms. The aim of this paper is to test such algorithms versus other algorithms from the literature near from ours, such as forward Euler approximation, external uniformization and a finite volume method from [C. Cocozza-Thivent, R. Eymard, Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme, ESAIM: M2AN 38(5) (2004) 853–875].     

A perturbed risk model with dependence between premium rates and claim sizes

This paper considers a dependent risk model with diffusion for the surplus of an insurer, in which a current premium rate will be adjusted after a claim occurs and the adjusted rate is determined by the amount of the claim. At the same time, the diffusion is changed correspondingly. Using Rouché’s theorem, we first derive the closed-form solution for the Laplace transform of the survival probability in the dependent risk model. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. For the exponential claim sizes, we present the explicit recursion expression for the survival probability, by which we can exactly solve the survival probability step-by-step. We also illustrate the influence of the model parameters in the dependent risk model on the survival probability by numerical examples.