保险费收取次数为泊松过程下的广义复合泊松风险模型

经典的破产模型是假定保险公司按单位时间常数速率收取保险费，盈余过程{R（t），t≥0中的S（f）=∑i=1^N（t）Y,为一复合泊松过程，本文将保费到达过程推广为一个Poisson过程，同时将S（t）推广为一个广义复合Poisson过程．针对此模型给出了盈余过程的一些性质，得到关于破产概率的一个定理．     

Two Ergodic Sample-Path Properties of the Poisson Process

For increasing sequences of real numbers we consider two types of asymptotic behavior that remind of the defining property of a (homogeneous) Poisson process according to which the numbers of points in disjoint intervals are independent and follow Poisson distributions with specified parameters. We prove that almost all paths of a Poisson process show this asymptotic behavior, and characterize the Poisson process by these properties. Further we discuss the connection to equidistribution notions.     

Mixed poisson processes and the probability of ruin

In the Poisson case there is a well known formula that relates the probability of ruin to the distribution function of aggregate claims. It is shown how this formula can be generalized to the mixed Poisson case.     

On Estimation of Poisson Intensity Functions

Under the presence of only one realization, we consider a computationally simple algorithm for estimating the intensity function of a Poisson process with exponential quadratic and cyclic of fixed frequency trends. We argue that the algorithm can successfully be used to estimate any Poisson intensity function provided that it has a parametric form.     

On the first birth and the last death in a generation in a multi-type Markov branching process

In a multi-type continuous time Markov branching process the asymptotic distribution of the first birth in and the last death (extinction) of the kth generation can be determined from the asymptotic behavior of the probability generating function of the vector Z(^k)(t), the size of the kth generation at time t, as t tends to zero or as t tends to infinity, respectively. Apart from an appropriate transformation of the time scale, for a large initial population the generations emerge according to an independent sum of compound multi-dimensional Poisson processes and become extinct like a vector of independent reversed Poisson processes. In the first birth case the results also hold for a multi-type Bellman-Harris process if the life span distributions are differentiable at zero.     

The number of lattice paths below a cyclically shifting boundary

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical “reflection” argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special “staircases.”     

On the Ruin Problem in a Markov-Modulated Risk Model

In this paper, we consider the compound Poisson risk model influenced by an external Markovian environment process, i.e. Markov-modulated compound Poisson model. The explicit Laplace transforms of Gerber–Shiu functions are obtained, while the explicit Gerber–Shiu functions are derived for the K _n -family claim size distributions in the two-states case.      

复合泊松过程的可加性

对复合泊松分布可加性的研究在许多的文献中都可以看到,本文首先应用特征函数的方法证明了复合泊松分布的可加性.以此为基础,结合对随机过程相关性质的讨论,证明了复合泊松过程也具有与复合泊松分布可加性相似的,某种意义上的可加性性质.     

On a relationship between Uspensky's theorem and poisson approximations

In this paper we show that Uspensky's expansion theorem for the Poisson approximation of the distribution of sums of independent Bernoulli random variables can be rewritten in terms of the Poisson convolution semigroup. This gives rise to exact evaluations and simple remainder term estimations for the deviations of the distributions in study with respect to various probability metrics, generalizing results of Shorgin (1977, Theory Probab. Appl., 22, 846–850). Finally, we compare the sharpness of Poisson versus normal approximations.     

On the central limit theorem for the stationary Poisson process of compact sets

Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on ?^d, derived from these processes using Hausdorff and projection measures are studied. The central limit theorem is formulated in a way which enables comparison of the various estimators of the intensity of the produced random measures. Approximate confidence intervals for the intensity are constructed. Their use is demonstrated in an example of length intensity estimation for the segment processes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Ramanujan equation connected with the median of the gamma distribution

In this paper, we consider the sequence solving the Ramanujan equation

The three main achievements are the following. We introduce a continuous-time extension of and show its close connections with the medians of the distributions and the Charlier polynomials. We give upper and lower bounds for both and , in particular for , which are sharper than other known estimates. Finally, we show (and at the same time complete) two conjectures by Chen and Rubin referring to the sequence of medians .

     

On risk processes with the Markov property and with independent increments

We study the influence on the underlying counting process of the Markov property and of the property of independent increments for a risk process.     

Computation of the Optimal Policy for the Control of a Compound Immigration Process through Total Catastrophes

In this paper we consider a Markov decision model introduced by Economou (2003), in which it was proved that the optimal policy in the problem of controlling a compound immigration process through total catastrophes is of control-limit type. We show that the average cost of a control-limit policy is unimodal as a function of the critical point. This result enables us to design very efficient algorithms for the computation of the optimal policy as the bisection procedure and a special-purpose policy iteration algorithm that operates on the class of control-limit policies.AMS 2000 Subject Classification: Primary 9OC40; Secondary 6OJ25     

Empirical Polygon Simulation and Central Limit Theorems for the Homogenous Poisson Line Process

For the Poisson line process the empirical polygon is defined thanks to ergodicity and laws of large numbers for some characteristics, such as the number of edges, the perimeter, the area, etc... One also has, still thanks to the ergodicity of the Poisson line process, a canonical relation between this empirical polygon and the polygon containing a given point. The aim of this paper is to provide numerical simulations for both methods: in a previous paper (Paroux, Advances in Applied Probability, 30:640–656, 1998) one of the authors proved central limit theorems for some geometrical quantities associated with this empirical Poisson polygon, we provide numerical simulations for this phenomenon which generates billions of polygons at a small computational cost. We also give another direct simulation of the polygon containing the origin, which enables us to give further values for empirical moments of some geometrical quantities than the ones that were known or computed in the litterature. This work was partially supported by the PSMN at ENS-Lyon.     

Skorohod representation on a given probability space

Let $(\Omega,\mathcal{A},P)Let be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and an arbitrary map, n = 1,2,.... If μ is tight and X _n converges in distribution to μ (in Hoffmann–J?rgensen’s sense), then X∼μ for some S-valued random variable X on . If, in addition, the X _n are measurable and tight, there are S-valued random variables and X, defined on , such that , X∼μ, and a.s. for some subsequence (n _k ). Further, a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X _n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to for some H⊂Ω with P ^*(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken , for some H⊂ (0,1) with outer Lebesgue measure 1, where is the Borel σ-field on (0,1) and m _H the only extension of Lebesgue measure such that m _H (H) = 1. In order to prove the previous results, it is also shown that, if X _n converges in distribution to a separable limit, then X _{n k} converges stably for some subsequence (n _k ).      

Characteristics of queueing systems observed at events and the connection between stochastic intensity and palm probability

This article explores the link between the concepts of stochastic intensity and Palm probability and gives a new proof and useful extensions to the so-called PASTA property of queueing theory.     

On a first-passage problem for a cumulative process with exponential decay

A first-passage problem for a cumulative process is investigated. The cumulative process is assumed to be generated by a Poisson process, and the amplitude generated by an event is assumed to decay exponentially. An integral equation for the probability density of the first-passage time until the total amplitude exceeds a pre-specified threshold level is derived. The Laplace transform of the probability density of the first-passage time is obtained explicity when each amplitude generated by an event is distributed exponentially. The mean first-passage times are given in a closed form and plotted versus the threshold level.     

On convergence of random walks generated by compound Cox processes to Lévy processes

A functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes to Lévy processes in the Skorokhod space. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with mixed normal distributions, and in particular, to stable Lévy processes.     

On multifunction transforms of probability measures

The paper briefly presents Topsøe's setup and gives two results based on that. A delta theorem for multifunctions improving that of King [2] is given. In particular, the contingent derivative may be a compact-valued instead of a single-valued multi-function. Stochastic optimization is also discussed. Some results are achieved for the limit distribution of optimal values as well as of optimal solutions.     

跳扩散模型下股价服从几何布朗运动的Esscher变换定价

考虑跳扩散模型下期权的Esscher变换定价,给出了Esscher变换下带跳的B-S矩生成函数和复合泊松过程下的矩生成函数,推导出跳扩散模型下期权的Esscher变换定价公式.