Stochastic Processes and their Applications最新文献

英文中文

Mixed orthogonality graphs for continuous-time stationary processes 连续时间静止过程的混合正交图

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-09 DOI: 10.1016/j.spa.2024.104501

Vicky Fasen-Hartmann, Lea Schenk

In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent characterisations are given for the different definitions, in particular by orthogonal projections. We then define two mixed graphs based on different definitions of Granger causality and contemporaneous correlation, the (mixed) orthogonality graph and the local (mixed) orthogonality graph. In these graphs, the components of the process are represented by vertices, directed edges between the vertices visualise Granger causal influences and undirected edges visualise contemporaneous correlation between the component processes. Further, we introduce various notions of Markov properties in analogy to Eichler (2012), which relate paths in the graphs to different dependence structures of subprocesses, and we derive sufficient criteria for the (local) orthogonality graph to satisfy them. Finally, as an example, for the popular multivariate continuous-time AR (MCAR) processes, we explicitly characterise the edges in the (local) orthogonality graph by the model parameters.

在本文中，我们为多变量静态连续时间过程引入了不同的格兰杰因果关系和同期相关性概念，以模拟各组成过程之间的不同依赖关系。本文给出了不同定义的几种等效特征，特别是正交投影。然后，我们根据格兰杰因果关系和同期相关性的不同定义定义了两种混合图，即（混合）正交图和局部（混合）正交图。在这些图中，流程的各组成部分由顶点表示，顶点之间的有向边表示格兰杰因果影响，无向边表示各组成部分流程之间的同期相关性。此外，我们还引入了与 Eichler（2012 年）类似的马尔可夫特性的各种概念，这些概念将图中的路径与子过程的不同依赖结构联系起来，并推导出（局部）正交图满足这些概念的充分标准。最后，以流行的多变量连续时间自回归（MCAR）过程为例，我们通过模型参数明确描述了（局部）正交图中的边。

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引用次数: 0

On weak and strong solutions of time inhomogeneous Itô’s equations with VMO diffusion and Morrey drift 关于具有 VMO 扩散和莫雷漂移的时间不均匀伊托方程的弱解和强解

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-09 DOI: 10.1016/j.spa.2024.104505

N.V. Krylov

We prove the existence of weak solutions of Itô’s stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are also discussed. Our results are new even if the drift term vanishes.

我们证明了具有不规则扩散和漂移项的莫雷空间伊托随机时间相关方程的弱解的存在性。我们还讨论了弱唯一性（一般是有条件的）和与强解有关的猜想。即使漂移项消失，我们的结果也是新的。

引用次数: 0

Well-Posedness of the generalised Dean–Kawasaki Equation with correlated noise on bounded domains 有界域上具有相关噪声的广义迪安-川崎方程的良好拟合性

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-09 DOI: 10.1016/j.spa.2024.104503

Shyam Popat

In this paper, we extend the notion of stochastic kinetic solutions introduced in Fehrman and Gess (2024) to establish the well-posedness of stochastic kinetic solutions of generalised Dean–Kawasaki equations with correlated noise on bounded,

C^{2}

-domains with Dirichlet boundary conditions. The results apply to a wide class of non-negative boundary data, which is based on certain a priori estimates for the solutions, that encompasses all non-negative constant functions including zero and all smooth functions bounded away from zero.

在本文中，我们扩展了 Fehrman 和 Gess (2024) 中引入的随机动力学解的概念，以建立具有相关噪声的广义 Dean-Kawasaki 方程的随机动力学解在有界、C2 域和 Dirichlet 边界条件上的良好提出性。这些结果适用于一类广泛的非负边界数据，它基于对解的某些先验估计，包括所有非负常数函数（包括零）和所有离零有界的平滑函数。

引用次数: 0

Dual process in the two-parameter Poisson–Dirichlet diffusion 双参数泊松-狄利克特扩散中的双重过程

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-05 DOI: 10.1016/j.spa.2024.104500

Robert C. Griffiths , Matteo Ruggiero , Dario Spanò , Youzhou Zhou

The two-parameter Poisson–Dirichlet diffusion takes values in the infinite ordered simplex and extends the celebrated infinitely-many-neutral-alleles model, having a two-parameter Poisson–Dirichlet stationary distribution. Here we identify a dual process for this diffusion and obtain its transition probabilities. The dual is shown to be given by Kingman’s coalescent with mutation, conditional on a given configuration of leaves. Interestingly, the dual depends on the additional parameter of the stationary distribution only through the test functions and not through the transition rates. After discussing the sampling probabilities of a two-parameter Poisson–Dirichlet partition drawn conditionally on another partition, we use these notions together with the dual process to derive the transition density of the diffusion. Our derivation provides a new probabilistic proof of this result, leveraging on an extension of Pitman’s Pólya urn scheme, whereby the urn is split after a finite number of steps and two urns are run independently onwards. The proof strategy exemplifies the power of duality and could be exported to other models where a dual is available.

双参数泊松-狄利克特扩散在无限有序单纯形中取值，并扩展了著名的无限多中性等位基因模型，具有双参数泊松-狄利克特静态分布。在此，我们确定了这种扩散的对偶过程，并获得了其过渡概率。结果表明，对偶过程是以给定的叶子配置为条件，由带有突变的金曼聚合过程给出的。有趣的是，对偶过程只通过检验函数而不是转换率来依赖于静态分布的附加参数。在讨论了以另一个分区为条件得出的双参数泊松-德里克利特分区的采样概率后，我们利用这些概念和对偶过程推导出了扩散的过渡密度。我们的推导为这一结果提供了一个新的概率证明，它利用了皮特曼的波利亚瓮计划的扩展，即在有限步数后拆分瓮，然后两个瓮独立运行。该证明策略体现了对偶性的威力，可用于其他有对偶性的模型。

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引用次数: 0

Erratum to: “Statistical test for an urn model with random multidrawing and random addition” [Stochastic Process. Appl. 158 (2023) 342-360] 勘误："随机多抽签和随机添加的瓮模型统计检验》[《随机过程。应用》158 (2023) 342-360] 更正

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-05 DOI: 10.1016/j.spa.2024.104495

Irene Crimaldi , Pierre-Yves Louis , Ida G. Minelli

引用次数: 0

Parameter estimation and singularity of laws on the path space for SDEs driven by Rosenblatt processes 罗森布拉特过程驱动的 SDE 的参数估计和路径空间上的奇异规律

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-05 DOI: 10.1016/j.spa.2024.104499

Petr Čoupek, Pavel Kříž, Bohdan Maslowski

In this paper, we study parameter identification for solutions to (possibly non-linear) SDEs driven by additive Rosenblatt process and singularity of the induced laws on the path space. We propose a joint estimator for the drift parameter, diffusion intensity, and Hurst index that can be computed from discrete-time observations with a bounded time horizon and we prove its strong consistency under in-fill asymptotics with a fixed time horizon. As a consequence of this strong consistency, singularity of measures generated by the solutions with different drifts is shown. This results in the invalidity of a Girsanov-type theorem for Rosenblatt processes.

在本文中，我们研究了由加性罗森布拉特过程和路径空间上诱导规律的奇异性驱动的（可能是非线性）SDEs 解的参数识别。我们提出了一种漂移参数、扩散强度和赫斯特指数的联合估计器，该估计器可从有界时间跨度的离散时间观测结果中计算得出。由于这种强一致性，不同漂移的解所产生的度量具有奇异性。这导致罗森布拉特过程的吉尔萨诺夫型定理失效。

引用次数: 0

Weak convergence of continuous-state branching processes with large immigration 大量移民的连续状态分支过程的弱收敛性

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-09-30 DOI: 10.1016/j.spa.2024.104497

Clément Foucart , Linglong Yuan

Functional limit theorems are established for continuous-state branching processes with immigration (CBIs), where the reproduction laws have finite first moments and the immigration laws exhibit large tails. Different regimes of immigration are identified, leading to limiting processes that are either subordinators, CBIs, extremal processes, or extremal shot noise processes.

为有移民的连续状态分支过程（CBIs）建立了功能极限定理，在这些过程中，繁殖规律具有有限的第一矩，而移民规律表现出大尾。确定了不同的移民机制，从而得出了从属过程、CBIs、极值过程或极值射噪声过程等极限过程。

引用次数: 0

Geodesics cross any pattern in first-passage percolation without any moment assumption and with possibly infinite passage times 测地线在第一通道渗流中穿过任何模式，无需任何矩假设，且通道时间可能无限长

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-09-27 DOI: 10.1016/j.spa.2024.104496

Antonin Jacquet

In first-passage percolation, one places nonnegative i.i.d. random variables (

T (e)

) on the edges of

Z^{d}

. A geodesic is an optimal path for the passage times

T (e)

. Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translate of this pattern. When we assume that the common distribution of the passage times satisfies a suitable moment assumption, it is shown in [Antonin Jacquet. Geodesics in first-passage percolation cross any pattern, arXiv:2204.02021, 2023] that, apart from an event with exponentially small probability, this number is linear in the distance between the extremities of the geodesic. This paper completes this study by showing that this result remains true when we consider distributions with an unbounded support without any moment assumption or distributions with possibly infinite passage times. The techniques of proof differ from the preceding article and rely on a notion of penalized geodesic.

在第一通道渗滤中，我们将非负 i.i.d. 随机变量 (T(e)) 放在 Zd 的边上。大地线是通过时间 T(e) 的最优路径。考虑时间环境的局部属性。我们称之为模式。我们将研究一条大地线穿过该模式平移的次数。当我们假设通过时间的共同分布满足一个合适的矩假设时，[Antonin Jacquet.第一通道渗流中的大地线穿过任何图案，arXiv:2204.02021, 2023]中表明，除了指数级小概率事件外，这一次数与大地线极点之间的距离呈线性关系。本文通过证明当我们考虑不带任何矩假设的无界支持分布或可能具有无限通过时间的分布时，这一结果仍然成立，从而完成了这一研究。证明技术与前文不同，它依赖于惩罚性大地线的概念。

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引用次数: 0

A class of processes defined in the white noise space through generalized fractional operators 通过广义分数算子在白噪声空间定义的一类过程

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-09-20 DOI: 10.1016/j.spa.2024.104494

Luisa Beghin , Lorenzo Cristofaro , Yuliya Mishura

The generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this class of processes (such as continuity, occupation density, variance asymptotics and persistence) according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation.

最近，根据曼德尔布罗特-范内斯表示法，通过黎曼-刘维尔类型的分数算子，对白噪声和灰噪声空间中的分数布朗运动进行了概括。我们的目的是通过更一般的分数导数和积分来扩展这一构造，我们将其定义为傅里叶乘法算子，然后通过伯恩斯坦函数将其特殊化。更准确地说，我们引入了一类内核驱动过程，作为特例，它包含了文献中的许多模型，包括分数布朗运动、节制分数布朗运动、奥恩斯坦-乌伦贝克过程。与前几种模型相比，我们的模型具有更大的通用性，因此具有更高的灵活性和更广泛的适用性。在此，我们根据算子的傅里叶符号或所选伯恩斯坦函数所满足的条件，推导出这一类过程的一些性质（如连续性、占据密度、方差渐近性和持久性）。另一方面，这些过程如果分别通过导数或积分型算子获得，无论其定义中使用的核如何，都会被证明显示出短程或长程依赖性。最后，通过这种结构，我们可以定义相应的噪声，并求解朗之文积分方程。

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引用次数: 0

A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens–Pitman model 高斯波动的鞅方法和埃文斯-皮特曼模型的迭代对数定律

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-09-18 DOI: 10.1016/j.spa.2024.104493

Bernard Bercu , Stefano Favaro

The Ewens–Pitman model refers to a distribution for random partitions of $[n] = {1, \dots, n}$ , which is indexed by a pair of parameters $α \in [0, 1)$ and $θ > - α$ , with $α = 0$ corresponding to the Ewens model in population genetics. The large $n$ asymptotic properties of the Ewens–Pitman model have been the subject of numerous studies, with the focus being on the number $K_{n}$ of partition sets and the number $K_{r, n}$ of partition subsets of size $r$ , for $r = 1, \dots, n$ . While for $α = 0$ asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for $α \in (0, 1)$ only almost-sure convergences are available, with the proof for $K_{r, n}$ being given only as a sketch. In this paper, we make use of martingales to develop a unified and comprehensive treatment of the large $n$ asymptotic behaviours of $K_{n}$ and $K_{r, n}$ for $α \in (0, 1)$ , providing alternative, and rigorous, proofs of the almost-sure convergences of $K_{n}$ and $K_{r, n}$ , and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for $K_{n}$ and $K_{r, n}$ .

Ewens-Pitman模型指的是[n]={1,...,n}的随机分区分布，它由一对参数α∈[0,1)和θ>-α索引，其中α=0对应于种群遗传学中的Ewens模型。关于 Ewens-Pitman 模型的大 n 渐近特性，已有许多研究，重点是 r=1,...n 时的分割集数 Kn 和大小为 r 的分割子集数 Kr,n。对于 α=0 已有几乎确定收敛和高斯波动的渐近结果，而对于 α∈(0,1)，只有几乎确定收敛的结果，Kr,n 的证明仅给出了一个草图。在本文中，我们利用马氏定理对 α∈(0,1) 时 Kn 和 Kr,n 的大 n 渐近行为进行了统一而全面的处理，提供了 Kn 和 Kr,n 的几乎确定收敛性的替代和严格证明，并涵盖了高斯波动的差距。我们还得到了 Kn 和 Kr,n 的新的迭代对数定律。

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