Exponential Bounds for Ruin Probability in Two Moving Average Risk Models with Constant Interest Rate

The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest force. Exponential bounds for ruin probabilities of an infinite time horizon are derived by the martingale method.     

The gerber-shiu expected discounted penalty function for Lévy insurance risk processes

In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity.     

Tail behavior of supremum of a random walk when Cramér’s condition fails

We investigate tail behavior of the supremum of a random walk in the case that Cramer＇s condition fails, namely, the intermediate case and the heavy-tailed ease. When the integrated distribution of the increment of the random walk belongs to the intersection of exponential distribution class and O-subexponential distribution class, under some other suitable conditions, we obtain some asymptotic estimates for the tail probability of the supremum and prove that the distribution of the supremum also belongs to the same distribution class. The obtained results generalize some corresponding results of N. Veraverbeke. Finally, these results are applied to renewal risk model, and asymptotic estimates for the ruin probability are presented.     

Large deviations for generalized compound Poisson risk models and its bankruptcy moments

We extend the classical compound Poisson risk model to the case where the premium income process, based on a Poisson process, is no longer a linear function. For this more realistic risk model, Lundberg type limiting results on the finite time ruin probabilities are derived. Asymptotic behaviour of the tail probabilities of the claim surplus process is also investigated.     

On the Effects of Risk Pooling in Supply Chain Management： Review and Extensions

The primary challenge in supply chain management （SCM） is matching supply with uncertain demand. Risk pooling is an efficient and promising strategy to meet this challenge by reducing the underlying demand uncertainty through aggregation. The main focus of this paper is to analyze the effects of risk pooling under different supply chain settings. There are two main contributions. First, we propose a mathematical framework which serves the multi-purpose of （1） unifying existing models on risk pooling in the literature, （2） providing new facets and insights of understanding existing results on risk pooling, and （3） setting up new ground for extending existing models and results. Second; we investigate one interesting effect of risk pooling, namely, the decreasing marginal return （or supermodularity）. We show that there are decreasing marginal returns in risk pooling practices under certain conditions, specifically when the demand is independent and identically distributed （I.I.D.） and normally distributed.     

The survival probability in finite time period in fully discrete risk model

The probabilities of the following events are first discussed in this paper: the insurance company survives to any fixed time k and the surplus at time k equals x≥1. The formulas for calculating such probabilities are deduced through analytical and probabilistic arguments respectively. Finally, other probability laws relating to risk are determined based on the probabilities mentioned above.     

Nonlinear autoregressive models with heavy-tailed innovation

In this paper, we discuss the relationship between the stationary marginal tail probability and the innovation's tail probability of nonlinear autoregressive models. We show that under certain conditions that ensure the stationarity and ergodicity, one dimension stationary marginal distribution has the heavy-tailed probability property with the same index as that of the innovation's tail probability.     

On tail behavior of nonlinear autoregressive functional conditional heteroscedastic model with heavy-tailed innovations

We study the tail probability of the stationary distribution of nonparametric non- linear autoregressive functional conditional heteroscedastic (NARFCH) model with heavy- tailed innovations.Our result shows that the tail of the stationary marginal distribution of an NARFCH series is heavily dependent on its conditional variance.When the innovations are heavy-tailed,the tail of the stationary marginal distribution of the series will become heavier or thinner than that of its innovations.We give some specific formulas to show how the increment or decrement of tail heaviness depends on the assumption on the con- ditional variance function.Some examples are given.     

The Application of Kernel Smoothing to Time Series Data

There are already a lot of models to fit a set of stationary time series, such as AR, MA, and ARMA models. For the non-stationary data, an ARIMA or seasonal ARIMA models can be used to fit the given data. Moreover, there are also many statistical softwares that can be used to build a stationary or non-stationary time series model for a given set of time series data, such as SAS, SPLUS, etc. However, some statistical softwares wouldn＇t work well for small samples with or without missing data, especially for small time series data with seasonal trend. A nonparametric smoothing technique to build a forecasting model for a given small seasonal time series data is carried out in this paper. And then, both the method provided in this paper and that in SAS package are applied to the modeling of international airline passengers data respectively, the comparisons between the two methods are done afterwards. The results of the comparison show us the method provided in this paper has superiority over SAS＇s method.     

ON THE RENEWAL RISK MODEL WITH INTEREST AND DIVIDEND

We consider that the reserve of an insurance company follows a renewal risk process with interest and dividend. For this risk process, we derive integral equations and exact infinite series expressions for the Cerber-Shiu discounted penalty function. Then we give lower and upper bounds for the ruin probability. Finally, we present exact expressions for the ruin probability in a special case of renewal risk processes.     

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.     

Uniform tail asymptotics for the aggregate claims with stochastic discount in the renewal risk models

Considering an insurer who is allowed to make risk-free and risky investments, as in Tang et al.(2010), the price process of the investment portfolio is described as a geometric L′evy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of extended regular variation, we obtain an asymptotically equivalent formula which holds uniformly for all time horizons, and furthermore, the same asymptotic formula holds for the finite-time ruin probabilities. The results extend the works of Tang et al.(2010).     

具有Vasicek利率的风险过程的破产概率的分解(英文)

In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities： the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.     

Upper tail probabilities of integrated Brownian motions

We obtain new upper tail probabilities of m-times integrated Brownian motions under the uniform norm and the Lp norm. For the uniform norm, Talagrand’s approach is used, while for the Lp norm, Zolotare’s approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities(large ball probabilities) for general Gaussian random variables in Banach spaces. As applications,explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of m-times integrated Brownian motions are presented as well.     

Ruin probabilities with insurance and financial risks having an FGM dependence structure

We consider a discrete-time risk model,in which insurance risks and financial risks jointly follow a multivariate Farlie-Gumbel-Morgenstern distribution,and the insurance risks are regularly varying tailed.Explicit asymptotic formulae are obtained for finite-time and infinite-time ruin probabilities.Some numerical results are also presented to illustrate the accuracy of our asymptotic formulae.     

基于Copula函数TVaR和EVaR的风险评估和资金配置

In this paper,the expressions of tail value of risk(TVaR)and exponential tail value of risk(EVaR)for the total risk portfolio are given,which are splitted into two cases: the bivariate case and the multivariate case according to the number of the insurances.Then the risk contributions of the insurances portfolio and the credit portfolio are also obtained. Further more,for clarifying the above results,a numerical example is given.     

一类基于进入过程保险风险模型的破产概率(英文)

In this note,one kind of insurance risk models with the policies having multiple validity times are investigated.Explicit expressions for the ruin probabilities are obtained by using the martingale method.As a consequence,the obtained probability serves as an upper bound for the ruin probability of a newly developed entrance processes based risk model.     

The Theory of Finite Models without Equal Sign

In this paper, it is the first time ever to suggest that we study the model theory of all finite structures and to put the equal sign in the same situtation as the other relations. Using formulas of infinite lengths we obtain new theorems for the preservation of model extensions, submodels, model homomorphisms and inverse homomorphisms. These kinds of theorems were discussed in Chang and Keisler＇s Model Theory, systematically for general models, but Gurevich obtained some different theorems in this direction for finite models. In our paper the old theorems manage to survive in the finite model theory. There are some differences between into homomorphisms and onto homomorphisms in preservation theorems too. We also study reduced models and minimum models. The characterization sentence of a model is given, which derives a general result for any theory T to be equivalent to a set of existential-universal sentences. Some results about completeness and model completeness are also given.     

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.     

DURATION OF NEGATIVE SURPLUS FOR A TWO STATE MARKOV-MODULATED RISK MODEL

We consider a continuous time risk model based on a two state Markov process, in which after an exponentially distributed time, the claim frequency changes to a different level and can change back again in the same way. We derive the Laplace transform for the first passage time to surplus zero from a given negative surplus and for the duration of negative surplus. Closed-form expressions are given in the case of exponential individual claim. Finally, numerical results are provided to show how to estimate the moments of duration of negative surplus.