Demetrios P. Lyberopoulos, Nikolaos D. Macheras
Some martingale characterizations of compound mixed Poisson processes are proven, extending S. Watanabe's (1964) martingale characterization of Poisson processes as well as the main result of Lyberopoulos and Macheras (2012), concerning martingale characterizations of mixed Poisson processes.
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Stochastic Hall-magnetohydrodynamics equations on ${\mathbb{R}}^{3}$ with random forces expressed in terms of the time homogeneous Poisson random measures are considered. We prove the existence of a global martingale solution. The construction of a solution is based on the Fourier truncation method, stochastic compactness method and a version of the Skorokhod theorem for non-metric spaces adequate for Poisson type random fields.
Time-changed multiparameter Poisson processes and martingales
该研究提出多参数泊松过程（MPP），推导其分布性质、积分渐近分布，分析三类时间变换变体，并利用多参数鞅性质得到MPP及其变体的鞅刻画。
If a given aggregate process $S$ is a compound mixed Poisson process under a probability measure $P$, a characterization of all probability measures $Q$ on the domain of $P$, such that $P$ and $Q$ are progressively equivalent and $S$ remains a compound mixed Poisson process with improved properties, is provided. This result generalizes earlier work of Delbaen & Haezendonck (1989). Implications related to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.
The solution of Poisson's equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study two different representations for the solution in countable state space, one based on regenerative structure and the other based on an infinite sum of expectations. We also consider integrability and related uniqueness issues associated with solutions to Poisson's equation, and provide verifiable Lyapunov conditions to support our theory. Our key results include a central limit theorem and law of the iterated logari…
Let $\tilde{N}\_{t}$ be a standard compensated Poisson process on $[0,1]$. We prove a new characterization of anticipating integrals of the Skorohod type with respect to $\tilde{N}$, and use it to obtain several counterparts to well established properties of semimartingale stochastic integrals. In particular we show that, if the integrand is sufficiently regular, anticipating Skorohod integral processes with respect to $\tilde{N}$ admit a pointwise representation as usual Itô integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorohod integral proces…
Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
本文推广了基于广义多重傅里叶级数的任意重数迭代Ito随机积分展开方法，获得了关于鞅泊松测度和鞅的积分展开，证明了均方收敛性并给出贝塞尔函数系的展开实例。
Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process $S$ under a probability measure $P$ we characterize all probability measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively equivalent and $S$ remains a compound renewal process under $Q$. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.
We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.
A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core
该研究构造有限计数鞅核，证明泊松混沌中满足单位方差与有限四阶矩的随机变量与标准正态的柯尔莫哥洛夫距离界，消除相关假设并获得 Malliavin 导数的定量估计。
A branching Lévy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet $(σ^2,a,Λ)$, where the branching Lévy measure $Λ$ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic …
Prediction of Gaussian Volterra Processes with Compound Poisson Jumps
该研究针对带复合泊松跳的高斯Volterra过程，推导得到了其预测规律。
We consider a Poisson process $η$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $λ$ of $η$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $η$), which was previously known only in the special case, when $λ$ is the product of Lebesgue measure on $\R_+$ and a $σ$-finite measure on another space $\BX$. Our proof is new and based on only a few basic properties of Poisson p…
We construct solutions to the randomly-forced Navier--Stokes--Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of [4] and satisfy a renormalized form of the continuity equation in the sense…
A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasin…
We show that the empirical distribution of the eigenvalues of the sample covariance matrix of certain random vectors (not necessarily independent entries) with bounded marginal $L^{4}$ norms converges weakly to a compound free Poisson distribution.
One major obstacle in applications of Stein's method for compound Poisson approximation is the availability of so-called magic factors (bounds on the solution of the Stein equation) with favourable dependence on the parameters of the approximating compound Poisson random variable. In general, the best such bounds have an exponential dependence on these parameters, though in certain situations better bounds are available. In this paper, we extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Po…
Multidimensional compound Poisson approximations for symmetric distributions
本文研究了对称分布独立同分布整数向量和的分布，采用伴随复合泊松分布和第二种阶Hipp型带符号复合泊松测度进行近似，并构建了Bergström型渐进行扩展，通过总变差度量估计近似精度，多数情况下精度为O(n^{-1})量级。
A New Compound Poisson Process and Its Fractional Versions
本研究提出由独立泊松随机变量加权和导出的新型复合泊松分布，给出其Bell多项式显式形式，并推广为复合泊松过程与时间分数阶复合泊松过程，证明后者的分布性质、推导相关矩并得到对应分数阶微分方程。
We are developing a new method for the analysis of queuing systems with heterogeneous in time and space compound (marked) Poisson input flow. The state space of the input flow is embedded in a higher-dimensional space with a homogeneous marked Poisson field on it. We prove limit theorems for partial sums of marks under the ordering of field points by coordinates.
Consider a surplus process which both of collected premium and payed claim size are two independent compound Poisson processes. This article derives two approximated formulas for the ruin probability of such surplus process, say double stochastic compound poisson process. More precisely, it provides two mixture exponential approximations for ruin probability of such double stochastic compound poisson process. Applications to long_term Bonus_Malus systems and a heavy-tiled claim size distribution have been given. Improvement of our findings compared to the Cramer- Lundberg upper bound has been …
In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.
Compound Poisson process approximation under $β$-mixing and stabilization
本文为$β$-混合点过程的稳定化统计量建立了Poisson与复合Poisson逼近并给出收敛速度，主要贡献包括扩展输入范围、给出Wasserstein距离下的具体界限，并通过随机几何示例验证了适用性。
In this article we determine the Laplace transforms of the main boundary functionals of the oscillating compound Poisson process. These are the first passage time of the level, the joint distribution of the first exit time from the interval and the value of the overshoot through the boundary. Under certain conditions we establish the asymptotic behaviour of the mentioned functionals.
We characterize the subexponential densities on $(0,\infty)$ for compound Poisson distributions on $[0,\infty)$ with absolutely continuous Lévy measures. As a corollary, we show that the class of all subexponential probability density functions on $\mathbb R_+$ is closed under generalized convolution roots of compound Poisson sums. Moreover, we give an application to the subexponential density on $(0,\infty)$ for the distribution of the supremum of a random walk.
Generalized specific entropy on Wiener space with application to Martingale Optimal Transport
本文提出了一种基于泊松跳跃近似的特定熵框架，用于连续时间鞅传输问题，解决了传统熵正则化在连续时间中的局限性，并引入了广义特定熵函数在维纳空间上的显式表达，提出了连续时间特定熵鞅最优传输问题SEMOT。
In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Lévy process consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, stopping time, the compactness method and the Jakubowski-Skorokhod theorem.
A characterization of mixed Poisson processes in terms of disintegrations is proven. As a consequence some further characterizations of such processes via claim interarrival processes, martingales and claim measures are obtained. Some revisions are required
In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (non-fractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.
Martingale Characterizations of Non-Homogeneous Counting Processes and Their Fractional Variants
本文提出非齐次计数过程的鞅特征及其分数变体的理论框架，通过稳定子 ordinators研究了其时间变换变体的鞅特征。
In this article, the compound Poisson processes of order $k$ (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinator (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the results in the literature.
Fractional Compound Poisson Random Fields on Plane
本文研究平面上的复合泊松随机场及其分数变体，推导分布、微分方程和弱收敛结果，并引入空间与时空分数泊松随机场，讨论性质并给出时间变换表示。
Asymptotic Results for Spectrally Positive Compound Poisson Processes
本文研究谱正复合泊松过程远离零的有限游走的渐进行为，探讨其上下方部分长度和高度的渐近关系，揭示其尺度对称性。
Skellam compound Poisson approximation to the sums of symmetric Markov dependent random variables
研究对称马尔可夫相依三点随机变量之和的近似问题，用Skellam随机变量近似该和，并在局部、总变差和Wasserstein度量下估计精度，利用测度卷积性质完成证明。
The Ewens-Pitman sampling model (EP-SM) is a distribution for random partitions of the set $\{1,\ldots,n\}$, with $n\in\mathbb{N}$, which is index by real parameters $α$ and $θ$ such that either $α\in[0,1)$ and $θ>-α$, or $α<0$ and $θ=-mα$ for some $m\in\mathbb{N}$. For $α=0$ the EP-SM reduces to the celebrated Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalization of the LS-CPSM, which is referred to as the negative Binomial compound Poisson sampling m…
Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(ξ(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(ξ(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit.
The term "moderate deviations" is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about "non-central moderate deviations" when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.
We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 173-179) and Roos (J. Multivariate Anal. 69 (1999) 120-134) in combination with new smoothness inequa…
A characterization of equivalent martingale probability measures in a mixed renewal risk model with applications in Risk Theory
本文刻画了使复合混合更新过程在逐步等价测度下变为复合混合 Poisson 过程的所有概率测度，并应用于破产问题和保费计算原则。
On the principal eigenvalue for compound Poisson processes
本文研究复合泊松过程在有界开集上主特征值的显式表达，证明当跳跃密度径向对称且严格递减时球体是唯一极小化子，并指出非严格递减时唯一性失效。
We study the default risk in incomplete information. That means, we model the value of a firm by one Lévy process which is the sum of brownian motion with drift and compound Poisson process. This Lévy process can not be observed completely and we let an other process which representes the available information on the firm. We obtain an equation safisfied by the conditional density of the default time given the available information and closed form expression for the density.
Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes. For spectrally negative compound Poisson processes with positive drift, a new representation of the $q$-scale functions in terms of the characteristics of the process is derived. Moreover, similar representations of the derivatives and the primitives of the $q$-scale functions are presented. The obtained formulae for the derivatives allow for a complete exposure of the smoothness properties of the considered $q$-scale functions. Some explicit examples of $q$-scale functions are given for illust…
Some Compound Fractional Poisson Processes
该研究引入并分析三种复合Poisson过程的分数阶版本，证明其为广义计数过程的极限情形，推导了相关统计性质与一维分布的等价形式，丰富了分数阶随机过程的理论研究。
The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.
Fractional Generalizations of the Compound Poisson Process
该研究提出统一的广义分数阶复合泊松过程GFCPP，推导其分布、微分方程及鞅性质，证明多种已有分数阶过程为其特例，还研究了时间变换分类并给出样本路径仿真方法。
We establish, under the Cramer exponential moment condition in a neighbourhood of zero, the Extended Large Deviation Principle for the Random Walk and the Compound Poisson processes in the metric space $\V$ of functions of finite variation on $[0,\infty)$ with the modified Borovkov metric $\r(f,g)= \r_\B(\hat{f},\hat{g}) $, where $ \hat f(t)= f(t)/(1+t)$, $t\in \R$, and $\r_\B$ is the Borovkov metric. LDP in this space is "more precise" than that with the usual metric of uniform convergence on compacts.
We consider two fractional versions of a family of nonnegative integer valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli, the Poisson Inverse Gaussian and the Negative Binomial. We also define and study some more general fractional versions with two fractional parameters.
The aim of the present work is to provide a supplement to the authors' paper (2018). It is shown that our results on the approximation of distributions of sums of independent summands by the accompanying compound Poisson laws and the estimates of the proximity of sequential convolutions of multidimensional distributions on convex polyhedra may be almost automatically transferred to the infinite-dimensional case.
Integrals of some compound processes
该研究基于复合泊松过程理论构建通用框架，证明伯恩斯坦从属的时间变换性质，推导复合过程积分的显式表示等结果，为相关随机过程研究提供理论支撑。
Approximate Modeling for Supercritical Galton-Watson Branching Processes with Compound Poisson-Gamma Distribution
本文研究了超临界分支过程在平均子代分布接近 1 的渐进行为下的种群大小分布，证明其可近似为复合泊松-伽马分布，并通过实验验证了该模型在电子倍增器等级联乘法过程中的有效性。
Modified ruin probability for a Cramér-Lundberg model driven by a compound mixed Poisson process
研究改进的破产概率，探讨在重尾和轻尾情况下，改进与经典破产概率的等价性及端点原子结果和密度渐近性。
Measure Concentration for Compound Poisson Distributions
本文通过修正 Herbst 论证和对数 Sobolev 不等式，为非负整数上的复合泊松测度导出了新的集中界，并在缺乏有限指数矩时得到最优多项式衰减，同时简化了 Houdré 和 Wu 的早期证明。
We use the Stein-Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. Both the cases that the fixed graph is a simple graph and that it has multiple edges are treated. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to gr…
A conditional compound Poisson process approach to the sparse Erdős-Rényi random graphs: moderate deviations
提出条件复合Poisson过程表示稀疏Erdős-Rényi随机图的连通分量，揭示其相变与零程模型凝聚转变的联系，并在非临界状态下推导了中等偏差原理。
Weak Martingale Solutions of the Stochastic Schrödinger-Poisson-Landau-Lifshitz-Gilbert System
本文研究含连续噪声与小跳噪声的三维随机 SPLLG 系统，借助惩罚泛函、Faedo-Galerkin 逼近、随机紧性及三层逼近证明弱鞅解存在性，并将既有结果推广至随机与耦合情形。
We show that for all $ψ$-mixing shifts distributions of the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of all points are close either to Poisson or to compound Poisson distributions. We also describe completely the asymptotic behavior of these distributions.
This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson process, where the Poisson counting process is replaced by the Mittag-Leffler counting process also known as fractional Poisson process. This process is no longer Markovian and Lévy. However, several analytical results are available and some of them are discussed here. The functional…
Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation
本文构建带标记二项过程的Markov-Malliavin结构与混沌展开体系，将其应用于复合泊松近似的Chen-Stein方法重构及三叉树模型投资组合优化研究。
Extreme values are considered in samples with random size that has a mixed Poisson distribution being generated by a doubly stochastic Poisson process. We prove some inequalities providing bounds on the rate of convergence in limit theorems for the distributions of max-compound Cox processes.
The fractional non-homogeneous Poisson process was introduced by a time-change of the non-homogeneous Poisson process with the inverse $α$-stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe's theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
The problem of European-style option pricing in time-changed Lévy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed Lévy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock pric…
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