A local limit theorem for the probability of ruin

In this paper, we give a result on the local asymptotic behaviour of the probability of ruin in a continuous-time risk model in which the inter-claim times have an Erlang distribution and the individual claim sizes have a distribution that belongs to S(v) with v≥ 0, but where the Lundberg exponent of the underlying risk process does not exist.     

LAPLACE TRANSFORM OF THE SURVIVAL PROBABILITY UNDER SPARRE ANDERSEN MODEL

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the survival probability is well given when the claim amount distribution is Erlang distribution or mixed Erlang distribution. The expressions for moments of the time to ruin with the model above are given.     

On the Expected Present Value of Total Dividends in a Risk Model with Potentially Delayed Claims

In this paper, we consider a risk model in which two types of individual claims, main claims and by-claims, are defined. Every by-claim is induced by the main claim randomly and may be delayed for one time period with a certain probability. The dividend policy that certain amount of dividends will be paid as long as the surplus is greater than a constant dividend barrier is also introduced into this delayed claims risk model. By means of the probability generating functions, formulae for the expected present value of total dividend payments prior to ruin are obtained for discrete-type individual claims. Explicit expressions for the corresponding results are derived for K n claim amount distributions. Numerical illustrations are also given.     

ON THE RUIN FUNCTIONS FOR A CORRELATED AGGREGATE CLAIMS MODEL WITH POISSON AND ERLANG RISK PROCESSES

This article considers a risk model as in Yuen et al. (2002). Under this model the two claim number processes are correlated. Claim occurrence of both classes relate to Poisson and Erlang processes. The formulae is derived for the distribution of the surplus immediately before ruin, for the distribution of the surplus immediately after ruin and the joint distribution of the surplus immediately before and after ruin. The asymptotic property of these ruin functions is also investigated.     

Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments

In two previous papers,the first named author jointly with Florian Luca and Henryk Iwaniec,have studied the distribution modulo 1 of sequences which have linear growth and are mean values of multiplicative functions on the set of all the integers.In this note,we give a first result concerning sequences with linear growth associated to the mean values of multiplicative functions on a set of polynomial values,proving the density modulo 1 of the sequencem[∑((m2+1))(m2+1)(m≤n)]n.This result is but an illustration of the theme which is currently being developed in the PhD thesis of the second named author.     

On the Distributions of Two Classes of Multiple Dependent Aggregate Claims

In this paper we examine two classes of correlated aggregate claims distributions, with univariate claim counts and multivariate claim sizes. Firstly, we extend the results of Hesselager [ASTIN Bulletin, 24： 19-32（1994）] and Wang ＆ Sobrero＇s [ASTIN Bulletin, 24：161-166 （1994）] concerning recursions for compound distributions to a multivariate situation where each claim event generates a random vector. Then we give a multivariate continuous version of recursive algorithm for calculating a family of compound distribution. Especially, to some extent, we obtain a continuous version of the corresponding results in Sundt [ASTIN Bulletin, 29：29-45 （1999）] and Ambagaspitiya [Insurance： Mathematics and Economics, 24：301-308 （1999）]. Finally, we give an example and show how to use the algorithm for aggregate claim distribution of first class to compute recursively the compound distribution.     

Algebraic convergence for discrete-time ergodic markov chains

This paper studies the l-ergodicity for discrete-time recurrent Markov chains.It proves that thel-order deviation matrix exists and is finite if and only if the chain is(l+2)-ergodic,and then the algebraicdecay rates of the n-step transition probability to the stationary distribution are obtained.The criteria forl-ergodicity are given in terms of existence of solution to an equation.The main results are,illustrated by someexamples.     

Optimal models with maximizing probability of first achieving target value in the preceding stages

Decision makers often face the need of performance guarantee with some sufficiently high probability. Such problems can be modelled using a discrete time Markov decision process (MDP) with a probability criterion for the first achieving target value. The objective is to find a policy that maximizes the probability of the total discounted reward exceeding a target value in the preceding stages. We show that our formulation cannot be described by former models with standard criteria. We provide the properties of the objective functions, optimal value functions and optimal policies. An algorithm for computing the optimal policies for the finite horizon case is given. In this stochastic stopping model, we prove that there exists an optimal deterministic and stationary policy and the optimality equation has a unique solution. Using perturbation analysis, we approximate general models and prove the existence of e-optimal policy for finite state space. We give an example for the reliability of the satellite sy     

An Image Denoising-Deblurring Adaptive Variational Problem in BV Space

In this paper, we establish the existence and uniqueness of a BV solution to an initial boundary value problem for a nonlinear integro-differential equation, which is related to a denoising and deblurring variational model in image restoration. Several experiments relevant to the restoration model are performed and numerical results are discussed.     

ARCH(0,1)系数中位无偏估计分布的渐近展开(英文)

This paper is concerned with the distributional properties of a median unbiased estimator of ARCH(0,1)coefficient.The exact distribution of the estimator can be easily derived,however its practical calculations are too heavy to implement, even though the middle range of sample sizes.Since the estimator is shown to have asymptotic normality,asymptotic expansions for the distribution and the percentiles of the estimator are derived as the refinements.Accuracies of expansion formulas are evaluated numerically,and the results of which show that we can effectively use the expansion as a fine approximation of the distribution with rapid calculations.Derived expansion are applied to testing hypothesis of stationarity,and an implementation for a real data set is illustrated.     

Sparre Andersen风险模型的破产问题(英)

本文主要研究了一类Sparre Andersen模型,其索赔时间间隔的分布为指数分布与Erlang(n) 分布的混合.得到了当初始资金u趋于无穷大时,破产概率ψ(u)的确切表达式和渐近表达式.     

Approximation of the Ultimate Ruin Probability in the Classical Risk Model Using Erlang Mixtures

In this paper, we approximate the ultimate ruin probability in the Cramér-Lundberg risk model when claim sizes have an arbitrary continuous distribution. We propose two approximation methods, based on Erlang Mixtures, which can be used for claim sizes distribution both light and heavy tailed. Additionally, using a continuous version of the empirical distribution, we develop a third approximation which can be used when the claim sizes distribution is unknown and paves the way for a statistical application. Numerical examples for the gamma, Weibull and truncated Pareto distributions are provided.     

On the absolute ruin problem in a Sparre Andersen risk model with constant interest

In this paper, we extend the work of Mitric and Sendova (2010), which considered the absolute ruin problem in a risk model with debit and credit interest, to renewal and non-renewal structures. Our first results apply to MAP processes, which we later restrict to the Sparre Andersen renewal risk model with interclaim times that are generalized Erlang (n) distributed and claim amounts following a Matrix-Exponential (ME) distribution (see for e.g. Asmussen and O’Cinneide (1997)). Under this scenario, we present a general methodology to analyze the Gerber-Shiu discounted penalty function defined at absolute ruin, as a solution of high-order linear differential equations with non-constant coefficients. Closed-form solutions for some absolute ruin related quantities in the generalized Erlang (2) case complement the results obtained under the classical risk model by Mitric and Sendova (2010).     

带干扰广义Elang(n)风险过程的破产前最大盈余

本文考虑了索赔时间间距为广义Erlang(n)分布的带干扰更新(Sparre Andersen)风险过程.所用的方法类似于Albrecher,et al.(2005),即将广义Erlang(n)随机变量分解成n个独立的指数随机变量的和.建立了破产前最大盈余所满足的积分-微分方程,讨论了索赔量分布为K<,m>分布时的特殊情形.     

On the expected discounted penalty function for the compound Poisson risk model with delayed claims

In this paper, we consider an extension to the compound Poisson risk model for which the occurrence of the claim may be delayed. Two kinds of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed with a certain probability. Both the expected discounted penalty functions with zero initial surplus and the Laplace transforms of the expected discounted penalty functions are obtained from an integro-differential equations system. We prove that the expected discounted penalty function satisfies a defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution, and an analytic expression for this quantity is given for when the claim amounts from both classes are exponentially distributed. Moreover, the closed form expressions for the ruin probability and the distribution function of the surplus before ruin are obtained. We prove that the ruin probability for this risk model decreases as the probability of the delay of by-claims increases. Finally, numerical results are also provided to illustrate the applicability of our main result and the impact of the delay of by-claims on the expected discounted penalty functions.     

具有二步保费的Erlang（2）风险模型

本文考虑了当索赔间隔时间为Erlang(2)分布且保费收取为二步保费过程的复合更新风险模型,推导出该模型的罚金折现期望值函数满足具有一定边界条件和积分微分方程,并解出该方程.特别地,当索赔额为指数分布时,利用所得结果给出了破产时间的Laplace变换及终积破产概率的解析解.     

带干扰phaes-type风险模型中的最大盈余及相关分布

本文考虑了索赔时间间距为phase-type分布时带干扰更新风险模型中的破产前最大盈余、破产后赤字的分布,建立了相应的积分-微分方程.最后,讨论了当索赔时间间距为Erlang(2)分布且索赔量满足指数分布时的特殊情形.     

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.