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.     

Optimal Conditional Estimation: Average Case Setting

We consider an estimation problem which appears in the identification of systems by means of restricted complexity models: find the optimal approximation to an element of a linear normed space (a system) based on noisy information, subject to the restriction that approximations (models) can be selected from a prescribed subspace M of the problem element space. In contrast to the worst-case optimization criterion, which may be pessimistic, in this paper the quality of an identification algorithm is measured by its local average performance. Two types of local average errors are considered: for a given information (measurement) y and for a given unknown element x, the latter in two versions. For a wide spectrum of norms in the measurement space, we define an optimal algorithm and give expressions for its average errors which show the dependence on information, information errors, unmodelled dynamics, and norm in the measurement space.     

On Semiconvexity Properties of Rotationally Invariant Functions in Two Dimensions

Let f be a function defined on the set M ^2×2 of all 2 by 2 matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function f can be represented as a function of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of f in terms of its representation      

索赔为稀疏过程的双复合 Poisson风险模型

研究了一类风险过程,其中保费收入为复合Poisson过程,而描述索赔发生的计数过程为保单到达过程的p-稀疏过程.给出了生存概率满足的积分方程及其在指数分布下的具体表达式,得到了破产概率满足的Lundberg不等式、最终破产概率及有限时间内破产概率的一个上界和生存概率的积分-微分方程,且通过数值例子,分析了初始准备金、保费收入、索赔支付及保单的平均索赔比例对保险公司破产概率的影响.     

Nonstationarity in statistical process control — issues,cases, ideas

Statistical process control (SPC) is a powerful framework that is used in many industries to decrease process variability and to pinpoint special cause variation. Although a broad range of techniques have been developed to do so, often the real‐life situation does not fully comply with the basic assumptions that are made in SPC resulting in poor results. One of the main violations against the assumptions is the fact that industrial processes rarely behave in a stationary manner — this is evidently the case for biological processes but is also an important issue when monitoring industrial processes. Besides, the ever increasing amount of data, with a clear shift towards multivariate and even multiway quality control, makes the classical univariate approach not feasible anymore. These two observations pose important challenges to statisticians to develop novel SPC algorithms that are broadly applicable in modern industries. In this contribution we discuss both issues and use two very different case studies to show the reader recent directions and developments in the SPC landscape. Copyright © 2011 John Wiley & Sons, Ltd.     

The largest table in Chinese restaurant processes

In this paper, we investigate the expectation of the size of the largest table in an (α, θ)-Chinese restaurant process by using and developing an idea originated in the work by Shepp, which discusses random permutation. This work was supported by National Natural Science Foundation of China (Grant No. 10671036) and the National Basic Research Program of China (Grant No. 2007CB814904)     

Spatial Point Process Models of Defensive Strategies: Detecting Changes

The study of stochastic processes can take many forms. Theoretical properties are important to ensure consistent model definition. Statistical inference on unknown parameters is equally important but can be difficult. This is principally because many of the standard assumptions for proving consistency and asymptotic normality of estimators involve independence and homogeneity. In the case where inference is concerned with detecting change in a spatial process from one time point to another, a statistical-computing approach can be rewarding. Regardless of the complexity of the stochastic process, if simulating from it is relatively easy, then detecting change is possible using a Monte Carlo approach. The methodology is applied in a military scenario, where a country’s defensive posture changes as a function of its perceived threat. For tactical-decision purposes, it is extremely important to know whether the country’s perceived threat level has changed.     

调整保险费率模型下的破产概率

本文主要讨论当保险费率按公司的盈余进行适当的调整时,如何求破产概率的问题．     

Backward Stochastic Differential Equation with Two Reflecting Barriers and Jumps

Abstract

In this paper, by using a penalization as well as a fixed point methods, we prove existence and uniqueness of the solution for the one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process.     

α－混合过程条件密度的估计及其性质

本文在{Xr,t∈N）是一个严平稳过程的假设下,用核估计的方法对未来状态XN＋T的条件密度进行估计．在假设{Xt,t∈N）是α-混合过程的情况下,讨论了过程有限维密度核估计的期望与方差,以及过程条件密度核估计的偏及均方误差．在一定条件下,证明了估计的弱收敛性．     

Continuous-Time Approximation of Short-Term Interest Rates in Generalized Ho-Lee Framework

In this paper, we study the weak convergence of short-term interest rate processes in multinomial (one-factor) and squared binomial (two-factor) generalizations of the Ho-Lee framework. We show that, under appropriate conditions on the rate of convergence of state probabilities and volatility parameter, in the one-factor case, the spot interest rate process converges to either Wiener process or superposition of Poisson processes. In the two-factor case, the limit process can have the form of the superposition of Wiener and Poisson components. The asymptotic results are proved under risk-neutral probability and local alternatives. Research is supported by the Lithuanian State Science and Studies Foundation, program “Mathematical Models of Lithuanian Economy for Forecasting of the Macroeconomic Processes” (registration No C-03004). __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 287–314, July–September, 2005.     

Estimating the intensity of a cyclic Poisson process in the presence of linear trend

We construct and investigate a consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process in the presence of linear trend. It is assumed that only a single realization of the Poisson process is observed in a bounded window. We prove that the proposed estimator is consistent when the size of the window indefinitely expands. The asymptotic bias, variance, and the mean-squared error of the proposed estimator are also computed. A simulation study shows that the first order asymptotic approximations to the bias and variance of the estimator are not accurate enough. Second order terms for bias and variance were derived in order to be able to predict the numerical results in the simulation. Bias reduction of our estimator is also proposed.     

Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain

LetC(A) be the convex hull generated by a Poisson point process in an unbounded convex setA. A representation ofAC(A) as the union of curvilinear triangles with independent areas is established. In the case whenA is a cone the properties of the representation are examined more completely. It is also indicated how to simulateC(A) directly without first simulating the process itself.     

Controlling jumps in correlated processes of Poisson counts

Processes of autocorrelated Poisson counts can often be modelled by a Poisson INAR(1) model, which proved to apply well to typical tasks of SPC. Statistical properties of this model are briefly reviewed. Based on these properties, we propose a new control chart: the combined jumps chart. It monitors the counts and jumps of a Poisson INAR(1) process simultaneously. As the bivariate process of counts and jumps is a homogeneous Markov chain, average run lengths (ARLs) can be computed exactly with the well‐known Markov chain approach. Based on an investigation of such ARLs, we derive design recommendations and show that a properly designed chart can be applied nearly universally. This is also demonstrated by a real‐data example from the insurance field. Copyright © 2008 John Wiley & Sons, Ltd.     

Limit laws for coverage in backbone-sensor networks

We study coverage in sensor networks having two types of nodes, namely, sensor nodes and backbone nodes. Each sensor is capable of transmitting information over relatively small distances. The backbone nodes collect information from the sensors. This information is processed and communicated over an ad hoc network formed by the backbone nodes, which are capable of transmitting over much larger distances. We consider two models of deployment for the sensor and backbone nodes. One is a Poisson–Poisson cluster model and the other a dependently thinned Poisson point process. We deduce limit laws for functionals of vacancy in both models using properties of association for random measures.     

膨胀的Poisson过程及其在金融中的应用

本文定义了一种增量不独立的纯跳过程,称为膨胀的Poisson过程.采用了一个n跳过程来描述股票市场价格运动的规律,并构造了一个货币市场投资组合使得它的市场价值在指定时刻与股票价格相等,且该投资组合的收益被分解成为一个确定性的项和一个膨胀的Poisson项之和.证明了投资投票市场风险大于投资货币市场风险.     

On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process  = {τ₀,τ₁,…} marked by a multivariate random walk (,) and its behavior about fixed levels to be crossed by one of the components of (,). We derive the joint distribution of first passage time τ_ρ, pre-exit time τ_ρ?1 (i.e., the instant one phase prior to the first passage time), and the respective values of (,) at τ_ρ and τ_ρ?1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk ((Π),) embedded in Π over . Processes like these can model various phenomena including stock market and option trading.

One of the central issues in the investigation of ((Π),) is to obtain the information about Π at any moment of time in random vicinities of τ_ρ and τ_ρ?1 previously available only upon . The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.     

Incorporating historical information in testing for a trend in Poisson means

An exact conditional test is developed for testing a trend in Poisson means when the historical control information is incorporated into the concurrent control data. An asymptotic conditional test is also developed as an alternative to the Tarone test. Asymptotic gains by the incorporation of the historical information is evaluated.     

Besov空间中双参数Volterra型重分数过程的弱逼近

In this paper, we prove that two-parameter Volterra multifractional process can be approximated in law in the topology of the anisotropic Besov spaces by the family of processes{B_n(s,t)},n∈N defined by B_n(s,t)=∫_0~s ∫_0~tk_(a(s))(s,u)K_(β(t))(t,u)θ_(n(u,v))dudv,here {θ_n(u, v)}n∈N is a family of processes, converging in law to a Brownian sheet as n→∞,based on the well known Donsker's theorem.