Stochastic Processes and their Applications最新文献

英文中文

Sticky diffusions on star graphs: Characterization and Itô formula 星图上的粘性扩散：表征和Itô公式

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-16 DOI: 10.1016/j.spa.2025.104795

Jules Berry , Fausto Colantoni

In this paper, we investigate continuous diffusions on star graphs with sticky behaviour at the vertex. These are Markov processes with continuous paths having a positive occupation time at the vertex. We characterize the sticky diffusions as time changed nonsticky diffusions by adapting the classical technique of Itô and McKean. We prove a form of Itô formula, also known as Freidlin–Sheu formula, for this type of process. As an intermediate step, we also obtain a stochastic differential equation satisfied by the radial component of the process. These results generalize those already known for sticky diffusions on a half-line and skew sticky diffusions on the real line.

本文研究了顶点具有粘性的星图上的连续扩散问题。这些是具有连续路径的马尔可夫过程在顶点处占用时间为正。我们采用Itô和McKean的经典方法将粘性扩散表征为随时间变化的非粘性扩散。对于这类过程，我们证明了Itô公式的一种形式，也称为Freidlin-Sheu公式。作为中间步骤，我们还得到了过程径向分量所满足的随机微分方程。这些结果推广了已知的半线上的粘性扩散和实线上的偏粘性扩散。

引用次数: 0

Fluid limits for interacting queues in sparse dynamic graphs 稀疏动态图中交互队列的流体限制

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-16 DOI: 10.1016/j.spa.2025.104794

Diego Goldsztajn , Sem C. Borst , Johan S.H. van Leeuwaarden

Consider a network of

n

single-server queues where tasks arrive independently at each server at rate

λ_{n}

. The servers are connected by a graph that is resampled at rate

μ_{n}

in a way that is symmetric with respect to the servers, and each task is dispatched to the shortest queue in the graph neighborhood where it appears. We aim to gain insight in the impact of the dynamic network structure on the load balancing dynamics in terms of the occupancy process which describes the empirical distribution of the number of tasks across the servers. This process evolves on the underlying dynamic graph, and its dynamics depend on the number of tasks at each individual server and the neighborhood structure of the graph. We establish that this dependency disappears in the limit as

n \to \infty

when

λ_{n} / n \to λ

and

μ_{n} \to \infty

, and prove that the limit of the occupancy process is given by a system of differential equations that depends solely on

λ

and the limiting degree distribution of the graph. We further show that the stationary distribution of the occupancy process converges to an equilibrium of the differential equations, and derive properties of this equilibrium that reflect the impact of the degree distribution. Our focus is on truly sparse graphs where the maximum degree is uniformly bounded across

n

, which is natural in load balancing systems.

考虑一个由n个单服务器队列组成的网络，其中任务以λn的速率独立到达每个服务器。服务器通过一个图来连接，该图以μn的速率以一种相对于服务器对称的方式重新采样，并且每个任务被分配到它出现的图邻域中最短的队列中。我们的目标是深入了解动态网络结构对占用过程中负载平衡动态的影响，该过程描述了跨服务器的任务数量的经验分布。这个过程在底层动态图上发展，它的动态性取决于每个单独服务器上的任务数量和图的邻域结构。我们证明了当λn/n→λ和μn→∞时，占据过程的极限在n→∞处消失，并证明了占据过程的极限由一个完全依赖λ的微分方程组和图的极限度分布给出。我们进一步证明了占有过程的平稳分布收敛于微分方程的平衡，并推导了反映度分布影响的平衡的性质。我们的重点是真正的稀疏图，其中最大度均匀地跨越n，这在负载平衡系统中是很自然的。

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引用次数: 0

Scaling limit for small blocks in the Chinese restaurant process 中餐馆过程中小块的缩放限制

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-14 DOI: 10.1016/j.spa.2025.104793

Oleksii Galganov , Andrii Ilienko

The Chinese restaurant process is a basic sequential construction of consistent random partitions. We consider random point measures describing the composition of small blocks in such partitions and show that their scaling limit is given by the projective limit of certain inhomogeneous Poisson measures on cones of increasing dimension. This result makes it possible to derive classical and functional limit theorems in the Skorokhod topology for various characteristics of the Chinese restaurant process.

中餐馆的过程是一个基本的顺序建设，一致的随机分区。我们考虑了描述这些分区中小块组成的随机点测度，并证明了它们的尺度极限是由增加维数的锥上的某些非齐次泊松测度的投影极限给出的。这一结果使得在Skorokhod拓扑中推导出适合中国餐饮过程各种特征的经典极限定理和泛函极限定理成为可能。

引用次数: 0

Change of numeraire for weak martingale transport 弱鞅输运的数值变化

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-13 DOI: 10.1016/j.spa.2025.104779

Mathias Beiglböck , Gudmund Pammer , Lorenz Riess

Change of numeraire is a classical tool in mathematical finance. Campi–Laachir–Martini (Campi et al., 2017) established its applicability to martingale optimal transport. We note that the results of Campi et al. (2017) extend to the case of weak martingale transport. We apply this to shadow couplings (in the sense of Beiglböck and Juillet (2021)), continuous time martingale transport problems in the framework of Huesmann–Trevisan (Huesmann and Trevisan, 2019) and in particular to establish the correspondence of stretched Brownian motion with its geometric counterpart. From a mathematical finance perspective, the geometric (stretched) Brownian motion and the corresponding geometric Bass local volatility model are more natural, and via the change of numeraire transform the efficient and well-understood algorithm for the Bass local volatility model can be adapted to this geometric counterpart.

数值变换是数学金融中的一种经典工具。Campi - laachir - martini （Campi et al., 2017）建立了其对鞅最优运输的适用性。我们注意到Campi等人（2017）的结果扩展到弱鞅输运的情况。我们将其应用于阴影耦合（Beiglböck和juliet（2021）的意义上），Huesmann - Trevisan框架中的连续时间矩阵输移问题（Huesmann和Trevisan， 2019），特别是建立拉伸布朗运动与其几何对应的对应关系。从数学金融的角度来看，几何（拉伸）布朗运动和相应的几何Bass局部波动模型更自然，通过改变数值变换，Bass局部波动模型的高效且易于理解的算法可以适应于这种几何对应物。

引用次数: 0

Steady state and mixing of two run-and-tumble particles interacting through jamming and attractive forces 通过干扰和吸引力相互作用的两种滚跑粒子的稳定状态和混合

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-11 DOI: 10.1016/j.spa.2025.104791

Leo Hahn

We study the long-time behavior of two run-and-tumble particles on the real line subjected to an attractive interaction potential and jamming interactions, which prevent the particles from crossing. We provide the explicit invariant measure, a useful tool for studying clustering phenomena in out-of-equilibrium statistical mechanics, for different tumbling mechanisms and potentials. An important difference with invariant measures of equilibrium systems are Dirac masses on the boundary of the state space, due to the jamming interactions. Qualitative changes in the invariant measure depending on model parameters are also observed, suggesting, like a growing body of evidence, that run-and-tumble particle systems can be classified into close-to-equilibrium and strongly out-of-equilibrium models. We also study the relaxation properties of the system, which are linked to the timescale at which clustering emerges from an arbitrary initial configuration. When the interaction potential is linear, we show that the total variation distance to the invariant measure decays exponentially and provide sharp bounds on the decay rate. When the interaction potential is harmonic, we give quantitative exponential bounds in a Wasserstein-type distance.

研究了两粒子在吸引相互作用势和干扰相互作用下在实线上的长时间运动行为。对于不同的翻滚机制和势能，我们提供了显式不变测度，这是研究非平衡统计力学中聚类现象的一个有用工具。与平衡系统的不变测度的一个重要区别是，由于干扰相互作用，状态空间边界上的狄拉克质量。我们还观察到依赖于模型参数的不变测度的质变，这表明，像越来越多的证据一样，奔跑和翻滚的粒子系统可以分为接近平衡和强烈非平衡模型。我们还研究了系统的松弛特性，这与从任意初始配置出现聚类的时间尺度有关。当相互作用势为线性时，我们证明了到不变测度的总变化距离呈指数衰减，并给出了衰减率的明确界限。当相互作用势为调和时，我们给出了wasserstein型距离的定量指数界。

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引用次数: 0

Itô’s formula for the flow of measures of Poisson stochastic integrals and applications Itô的泊松随机积分测度流公式及其应用

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.spa.2025.104788

Thomas Cavallazzi

We prove Itô’s formula for the flow of measures associated with a jump process defined by a drift, an integral with respect to a Poisson random measure and with respect to the associated compensated Poisson random measure. We work in

P_{β} (R^{d})

, the space of probability measures on

R^{d}

having a finite moment of order

β \in (0, 2]

. As an application, we exhibit the backward Kolmogorov partial differential equation stated on

[0, T] \times P_{β} (R^{d})

associated with a McKean–Vlasov stochastic differential equation driven by a Poisson random measure. It describes the dynamics of the semigroup acting on functions defined on

P_{β} (R^{d})

associated with the McKean–Vlasov stochastic differential equation, under regularity assumptions on it. Finally, we use the semigroup and the backward Kolmogorov equation to prove new quantitative weak propagation of chaos results for a mean-field system of interacting Ornstein–Uhlenbeck processes driven by i.i.d.

α

-stable processes with

α \in (1, 2)

我们证明了Itô关于由漂移、泊松随机测度和相关补偿泊松随机测度的积分所定义的与跳跃过程相关的测度流的公式。我们在Pβ(Rd)中工作，在Rd上具有有限阶矩的概率测度空间β∈(0,2)。作为一个应用，我们展示了在[0，T]×Pβ(Rd)上表示的后向Kolmogorov偏微分方程与由泊松随机测度驱动的McKean-Vlasov随机微分方程相关联。在正则性假设下，描述了与McKean-Vlasov随机微分方程相关的半群作用于Pβ(Rd)上定义的函数的动力学。最后，我们利用半群和后向Kolmogorov方程证明了由α∈(1,2)的i.i.d α-稳定过程驱动的相互作用Ornstein-Uhlenbeck过程的平均场系统混沌结果的新定量弱传播。

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引用次数: 0

Long time fluctuations at critical parameter of Hopf’s bifurcation Hopf分岔临界参数的长时间波动

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-04 DOI: 10.1016/j.spa.2025.104785

M. Aleandri , P. Dai Pra

A dynamical system that undergoes a supercritical Hopf’s bifurcation is perturbed by a multiplicative Brownian motion that scales with a small parameter

ɛ

. The random fluctuations of the system at the critical point are studied when the dynamics starts near equilibrium, in the limit as

ɛ

goes to zero. Under a space–time scaling the system can be approximated by a 2-dimensional process lying on the center manifold of the Hopf’s bifurcation and a slow radial component together with a fast angular component are identified. Then the critical fluctuations are described by a “universal” stochastic differential equation whose coefficients are obtained taking the average with respect to the fast variable.

一个经历超临界霍普夫分岔的动力系统会受到一个乘性布朗运动的扰动，该运动的尺度为一个小参数。研究了系统在接近平衡状态时，在极限情况下，系统在临界点处的随机波动。在时空尺度下，系统可近似为Hopf分岔中心流形上的二维过程，并识别出慢速径向分量和快速角分量。然后用一个“通用”随机微分方程来描述临界波动，该方程的系数对快速变量取平均值。

引用次数: 0

Homeomorphism of the Revuz correspondence for finite energy integrals 有限能量积分的Revuz对应的同胚性

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-04 DOI: 10.1016/j.spa.2025.104787

Takumu Ooi

We provide necessary and sufficient conditions for the convergence of Revuz measures of finite energy integrals. More precisely, the Revuz map from the set of all smooth measures of finite energy integrals, equipped with the topology induced by the norm given by the sum of the Dirichlet form and the

L^{2} (m)

-norm, to the space of positive continuous additive functionals, equipped with the topology induced by the

L^{2} (P_{m + κ + ν_{0}})

-norm with the local uniform topology, is a homeomorphism, where

m

is the underlying measure,

κ

is the killing measure of a Dirichlet form and

ν_{0}

is an energy functional corresponding to the part of the process that continuously escapes to the cemetery point.

给出有限能量积分的Revuz测度收敛的充分必要条件。更准确地说，Revuz映射从有限能量积分的所有光滑测度的集合，具有由Dirichlet形式和L2(m)范数之和引起的拓扑，到由L2（Pm+κ+ν0）范数引起的拓扑与局部一致拓扑组成的正连续加性泛函空间，是一个同纯映射，其中m是底层测度，κ是狄利克雷形式的杀戮度量，ν0是一个能量泛函，对应于连续逃逸到墓地点的过程的一部分。

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引用次数: 0

Mokobodzki’s intervals: An approach to Dynkin games when value process is not a semimartingale Mokobodzki的间隔：当价值过程不是半鞅时，Dynkin游戏的一种方法

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-10-03 DOI: 10.1016/j.spa.2025.104786

Tomasz Klimsiak , Maurycy Rzymowski

We study Dynkin games governed by a nonlinear

E^{f}

-expectation on a finite interval

[0, T]

, with payoff càdlàg processes

L, U

of class (D) which are not imposed to satisfy (weak) Mokobodzki’s condition – the existence of a càdlàg semimartingale between the barriers. For that purpose we introduce the notion of Mokobodzki’s stochastic intervals

ℳ (θ)

(roughly speaking, maximal stochastic interval on which Mokobodzki’s condition is satisfied when starting from the stopping time

θ

) and the notion of reflected BSDEs without Mokobodzki’s condition. We prove an existence and uniqueness result for RBSDEs with driver

f

that is non-increasing with respect to the value variable (no restrictions on the growth) and Lipschitz continuous with respect to the control variable, and with data in

L^{1}

spaces. Next, we show numerous results on Dynkin games with most notable saying that the game is not played beyond

ℳ (θ)

, when starting from

θ

我们研究了有限区间[0，T]上由非线性ef -期望控制的Dynkin对策，该类(D)的收益càdlàg过程L，U不满足（弱）Mokobodzki条件-障碍之间存在càdlàg半鞅。为此，我们引入了Mokobodzki随机区间的概念（即从停止时间θ开始满足Mokobodzki条件的最大随机区间）和不满足Mokobodzki条件的反射BSDEs的概念。我们证明了一类RBSDEs的存在唯一性结果，该RBSDEs的驱动因子f对值变量不递增（对增长没有限制），对控制变量Lipschitz连续，且数据在L1空间中。接下来，我们展示了许多关于Dynkin游戏的结果，其中最值得注意的是，当从θ开始时，游戏不会超出tag （θ）。

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引用次数: 0

Limit theorems for functionals of linear processes in critical regions 临界区域线性过程泛函的极限定理

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-27 DOI: 10.1016/j.spa.2025.104784

Yudan Xiong , Fangjun Xu , Jinjiong Yu

Let

X = {X_{n} : n \in N}

be the linear process defined by

X_{n} = \sum_{j = 1}^{\infty} a_{j} ɛ_{n - j}

, where the coefficients

a_{j} = j^{- β} ℓ (j)

are constants with

β > 0

and

ℓ

a slowly varying function, and the innovations

{ɛ_{n}}_{n \in Z}

are i.i.d. random variables belonging to the domain of attraction of an

α

-stable law with

α \in (0, 2]

. Limit theorems for the partial sum

S_{[N t]} = \sum_{n = 1}^{[N t]} [K (X_{n}) - E K (X_{n})]

with proper measurable functions

K

have been extensively studied, except for two critical regions: I.

α \in (1, 2), β = 1

and II.

α β = 2, β \geq 1

. In this paper, we address these open scenarios and identify the asymptotic distributions of

S_{[N t]}

under mild conditions.

设X={Xn:n∈n}是由Xn=∑j=1∞aj æ n−j定义的线性过程，其中系数aj=j−β Z (j)是具有β>；0的常数，α∈(0,2)的缓变函数，创新项{æ n}n∈Z是属于α-稳定定律吸引域的i.d个随机变量。具有适当可测函数K的部分和S[Nt]=∑n=1[Nt][K(Xn)−EK(Xn)]的极限定理已经得到了广泛的研究，除了两个临界区域：I. α∈(1,2),β=1和II。αβ= 2,β≥1。在本文中，我们解决了这些开放的情况，并确定了S[Nt]在温和条件下的渐近分布。

引用次数: 0

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