Derived random measures

A derived random measure is constructed by integration of a random process with respect to a random measure independent of that process. Basic distributional properties, a continuity theorem, sample path properties, a strong law of large numbers, and a central limit theorem for derived random measures are established. Applications are given to compounding and thinning of point processes and the measure of a random set.     

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.     

Martingale Representation of Functionals of Lévy Processes

Abstract

The main focus of the paper is a Clark–Ocone–Haussman formula for Lévy processes. First a difference operator is defined via the Fock space representation of L ²(P), then from this definition a Clark–Ocone–Haussman type formula is derived. We also derive some explicit chaos expansions for some common functionals. Later we prove that the difference operator defined via the Fock space representation and the difference operator defined by Picard [Picard, J. Formules de dualitésur l'espace de Poisson. Ann. Inst. Henri Poincaré 1996, 32 (4), 509–548] are equal. Finally, we give an example of how the Clark–Ocone–Haussman formula can be used to solve a hedging problem in a financial market modelled by a Lévy process.     

Forward-backward stochastic differential equations with Brownian motion and poisson process

1.IntroductionLet(n,Y,{S}tZo,P)beastochasticbasissuchthatAscontainsallp-nullelementsofFand5 =nR .=h,t2o.Wesupposethatthefiltration{R}tZoisgeneratede>0bythefollowingtwOmutuallyindependentProcesses:(i)Ad-dbonsionalstandardBroedanmotion{Bt}tZo;(h)APoissonrandommeasureNonR xZ,whereZCFIisanonemptyopensetequippedwithitsBorelheldB(Z),withcompensatorN(dz,dt)=A(dz)dt,suchthatN(Ax[0,t])=(N--N)(Ax10,t])tZoisamartingaleforallAEB(Z)satisfyingA(A)相似文献   

Asymptotic behavior of a system of interacting nuclear-space-valued stochastic differential equations driven by Poisson random measures

In this paper we study a system of interacting stochastic differential equations taking values in duals of nuclear spaces driven by Poisson random measures. We also consider the McKean-Vlasov equation associated with the system. We show that under suitable conditions the system has a unique solution and the sequence of its empirical distributions converges to the solution of the McKean-Vlasov equation when the size of the system tends to infinity. The results are applied to the voltage potentials of a large system of neurons and the limiting distribution of the empirical measure is obtained.This research was supported by the National Science Foundation, the Air Force Office of Scientific Research under Grant No. F49620-92-J-0154, and the Army Research Office under Grant No DAAL03-92-G-0008.     

Position dependent and stochastic thinning of point processes

A short probabilistic proof of Kallenberg's theorem [2] on thinning of point processes is given. It is extended to the case where the probability of deletion of a point depends on the position of the point and is itself random. The proof also leads easily to a statement about the rate of convergence in Renyi's theorem on thinning a renewal process.     

泊松冲击下冷贮备可修系统的可靠性分析

本文研究了一类由有限个同质部件和一个修理工组成的冷贮备可修系统在随机冲击下的可靠性问题。假设冲击以泊松过程到达。当冲击到达时,它会独立地对系统中工作的部件产生影响,而不会对冷贮备部件产生影响。每次冲击的量都服从某一确定的分布,受冲击的部件以一定的概率发生故障,其故障概率是冲击量的函数,当工作的部件发生故障时,下一个冷贮备部件立即开始工作,当所有部件故障时,系统故障,故障部件按故障顺序进行修理,修理时间服从指数分布,故障部件能被修理如新。本文显式给出了系统首次故障前平均时间、稳态可用度、稳态故障频度等可靠性指标。     

Ordered thinnings of point processes and random measures

This is a study of thinnings of point processes and random measures on the real line that satisfy a weak law of large numbers. The thinning procedures have dependencies based on the order of the points or masses being thinned such that the thinned process is a composition of two random measures. It is shown that the thinned process (normalized by a certain function) converges in distribution if and only if the thinning process does. This result is used to characterize the convergence of thinned processes to infinitely divisible processes, such as a compound Poisson process, when the thinning is independent and nonhomogeneous, stationary, Markovian, or regenerative. Thinning by a sequence of independent identically distributed operations is also discussed. The results here contain Renyi's classical thinning theorem and many of its extensions.     

Generalized Modulated Random Measures and Their Potentials

Abstract

This article deals with a class of random measures formed of doubly stochastic marked random measures that assumes parameters in accordance with the evolution of some stochastic process, called a “modulator.” Throughout the paper, restrictions imposed on random measures (to be modulated) and the modulator are kept to a minimum. One of the objective of these studies are intensities and reward rates of modulated random measures that can play a significant role in stochastic control and optimization. Analytically tractable formulas for such functionals are obtained and examples and applications are discussed and treated in details.     

Limit of Solutions of a SDE with a Large Drift Driven by a Poisson Random Measure

We consider a sequence of {X _n} of R ^d-valued processes satisfying a stochastic differential equation driven by a Brownian motion and a compensated Poisson random measure, with _n ~ _n with a large drift. Let be a m-dimensional submanifold (m<d), where F vanishes. Then under some suitable growth conditions for _n ~ _n, and some conditions for F, we show that dist(X _n, )0 before it exits any given compact set, that is, the large drift term forces X _n close to . And if the coefficients converge to some continuous functions, any limit process must actually stay on and satisfy a certain stochastic differential equation driven by Brownian motion and white noise.     

Catalytic Discrete State Branching Models and Related Limit Theorems

Catalytic discrete state branching processes with immigration are defined as strong solutions of stochastic integral equations. We provide main limit theorems of those processes using different scalings. The class of limit processes of the theorems includes essentially all continuous state catalytic branching processes and spectrally positive regular affine processes.      

Poisson过程分解问题的一个注记

在研究Poisson过程分解问题时,现有文献的证明往往令人费解,本文主要运用极限理论,给出了一个简明易懂的证明.     

The explicit chaotic representation of the powers of increments of Lévy processes

This paper presents a computationally explicit formula of the chaotic representation property (CRP) for the powers of increments of a Lévy process. The formula can be used to obtain the integrands of the CRP in terms of orthogonal compensated power jump processes and the CRP in terms of Poisson random measures. Simulation results demonstrate that the performance of the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of the powers of increments of the underlying Lévy process using Taylor's expansion.     

Series expansions of multiple Lévy integrals

Multiple stochastic integrals with respect to an infinitely divisible symmetric random measure without a Gaussian component admit LePage-type representations by means of certain multiple random series.     

基于模糊随机变量的结构可靠性分析

研究了广义应力和广义强度同时具有模糊性和随机性时的结构可靠度计算问题,基于模糊随机变量和模糊随机事件的理论,建立了结构模糊可靠度的计算模型.最后通过一算例,验证了该方法的有效性和合理性.