Stochastic Processes and their Applications最新文献

英文中文

Some remarks on the effect of the Random Batch Method on phase transition 关于随机分批法对相变影响的一些评论

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-16 DOI: 10.1016/j.spa.2024.104498

Arnaud Guillin , Pierre Le Bris , Pierre Monmarché

In this article, we focus on two toy models : the Curie–Weiss model and the system of

N

particles in linear interactions in a double well confining potential. Both models, which have been extensively studied, describe a large system of particles with a mean-field limit that admits a phase transition. We are concerned with the numerical simulation of these particle systems. To deal with the quadratic complexity of the numerical scheme, corresponding to the computation of the

O (N^{2})

interactions per time step, the Random Batch Method (RBM) has been suggested. It consists in randomly (and uniformly) dividing the particles into batches of size

p > 1

, and computing the interactions only within each batch, thus reducing the numerical complexity to

O (N p)

per time step. The convergence of this numerical method has been proved in other works.

This work is motivated by the observation that the RBM, via the random constructions of batches, artificially adds noise to the particle system. The goal of this article is to study the effect of this added noise on the phase transition of the nonlinear limit, and more precisely we study the effective dynamics of the two models to show how a phase transition may still be observed with the RBM but at a lower critical temperature.

本文重点讨论两个玩具模型：居里-韦斯模型和双井约束势中线性相互作用的 N 粒子系统。这两个模型都已被广泛研究，它们描述了一个大型粒子系统，其平均场极限允许相变。我们关注的是这些粒子系统的数值模拟。为了解决数值方案的二次方复杂性（相当于每个时间步计算 O(N2) 次相互作用），我们提出了随机批处理方法（RBM）。它包括随机（均匀）地将粒子分成大小为 p>1 的批次，并只计算每个批次内的相互作用，从而将每个时间步的数值复杂度降低到 O(Np)。这一数值方法的收敛性已在其他著作中得到证明。这项工作的动机是观察到 RBM 通过批次的随机构造人为地增加了粒子系统的噪声。本文的目的是研究这种增加的噪声对非线性极限相变的影响，更准确地说，我们研究了这两种模型的有效动力学，以说明如何在较低的临界温度下仍然可以观察到 RBM 的相变。

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引用次数: 0

Stochastic representation for solutions of a system of coupled HJB-Isaacs equations with integral–differential operators 带积分微分算子的 HJB-Isaacs 耦合方程组解的随机表示法

IF 1.1 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2024-10-15 DOI: 10.1016/j.spa.2024.104502

Sheng Luo , Wenqiang Li , Xun Li , Qingmeng Wei

In this paper, we focus on the stochastic representation of a system of coupled Hamilton–Jacobi–Bellman–Isaacs (HJB–Isaacs (HJBI), for short) equations which is in fact a system of coupled Isaacs’ type integral-partial differential equation. For this, we introduce an associated zero-sum stochastic differential game, where the state process is described by a classical stochastic differential equation (SDE, for short) with jumps, and the cost functional of recursive type is defined by a new type of backward stochastic differential equation (BSDE, for short) with two Poisson random measures, whose wellposedness and a prior estimate as well as the comparison theorem are investigated for the first time. One of the Poisson random measures

μ

appearing in the SDE and the BSDE stems from the integral term of the HJBI equations; the other random measure in BSDE is introduced to link the coupling factor of the HJBI equations. We show through an extension of the dynamic programming principle that the lower value function of this game problem is the viscosity solution of the system of our coupled HJBI equations. The uniqueness of the viscosity solution is also obtained in a space of continuous functions satisfying certain growth condition. In addition, also the upper value function of the game is shown to be the solution of the associated system of coupled Isaacs’ type of integral-partial differential equations. As a byproduct, we obtain the existence of the value for the game problem under the well-known Isaacs’ condition.

在本文中，我们将重点研究耦合汉密尔顿-雅各比-贝尔曼-艾萨克斯（简称 HJB-艾萨克斯（HJBI））方程组的随机表示，该方程组实际上是一个耦合艾萨克斯型积分-部分微分方程组。为此，我们引入了一个相关的零和随机微分博弈，其中状态过程由一个带跳跃的经典随机微分方程（简称 SDE）描述，递归类型的代价函数由一个带有两个泊松随机度量的新型后向随机微分方程（简称 BSDE）定义。在 SDE 和 BSDE 中出现的泊松随机量之一 μ 源自 HJBI 方程的积分项；BSDE 中的另一个随机量是为了连接 HJBI 方程的耦合因子而引入的。我们通过对动态编程原理的扩展证明，该博弈问题的低值函数就是我们的耦合 HJBI 方程系统的粘性解。在满足一定增长条件的连续函数空间中，我们还得到了粘性解的唯一性。此外，还证明了博弈的上值函数是相关的耦合艾萨克式积分偏微分方程系的解。作为副产品，我们在著名的艾萨克斯条件下得到了博弈问题的存在值。

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引用次数: 0

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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