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 asymptotic behaviour of Lévy processes,Part I: Subexponential and exponential processes

We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.     

A consistent set of infinite-order probabilities

Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent.May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a probability of a probability of a probability, and so on, ad infinitum? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines.     

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本文考虑了常利力下带干扰的双复合Poisson风险过程, 借助微分和伊藤公式, 分别获得了无限时和有限时生存概率的积分微分方程. 当保费服从指数分布时, 得到了无限时生存概率的微分方程.     

Stationary tail probabilities in exponential server tandems with renewal arrivals

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths of the arrival and virtual service processes in order to get a large queue size at the second queue. We then use large deviations estimates of the probabilities of these paths, and solve a constrained convex optimization problem to find the most likely path leading to a large queue size. We find that the stationary queue length distribution at the second queue has an exponentially decaying tail, and obtain the exact rate of decay.Research supported in part by NSF grant NCR 88-57731 and the AT & T Foundation.     

A general importance sampling algorithm for estimating portfolio loss probabilities in linear factor models

This paper develops a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. We apply exponential tilts to (i) the distribution of the natural sufficient statistics of the systematic risk factors and (ii) conditional default probabilities, given the simulated values of the systematic risk factors, and select parameter values by minimizing the Kullback–Leibler divergence of the resulting parametric family from the ideal (zero-variance) importance density. Optimal parameter values are shown to satisfy intuitive moment-matching conditions, and the asymptotic behaviour of large portfolios is used to approximate the requisite moments. In a sense we generalize the algorithm of Glasserman and Li (2005) so that it can be applied in a wider variety of models. We show how to implement our algorithm in the $t$ copula model and compare its performance there to the algorithm developed by Chan and Kroese (2010). We find that our algorithm requires substantially less computational time (especially for large portfolios) but is slightly less accurate. Our algorithm can also be used to estimate more general risk measures, such as conditional tail expectations, whereas Chan and Kroese (2010) is specifically designed to estimate loss probabilities.     

Asymptotics of multivariate conditional risk measures for Gaussian risks

This paper investigates accurate approximations of marginal moment excess, marginal conditional tail moment and marginal moment shortfall for multivariate Gaussian system risks. Based on the dimension reduction property via the quadratic programming problem, the super-exponential and polynomial convergence speeds are specified. Two interesting questions involved in risk management are well addressed, namely the minimal additional risk capital injection to avoid infinite risk contagion and a sufficient and necessary condition to alternate the convergence speeds. Numerical study and typical examples are given to illustrate the efficiency of our findings. Due to the flexible moment order, additional applications may involve in risk management, including tail mean–variance portfolio and multivariate conditional risk measures of tail covariance, tail skewness with dependence and extremal risk contagion under consideration.     

Serial dependence in ARCH-models as measured by tail dependence coefficients

Serial dependence in non-linear time series cannot always be reliably quantified using linear autocorrelation. We do a detailed study of serial dependence in an ARCH(1) process from the point of view of the lower tail dependence coefficient and certain generalisations thereof. Our results are relevant for estimating probabilities of consecutive value-at-risk violations in GARCH models.