Stochastic Processes and their Applications最新文献

英文中文

Duality for some models of epidemic spreading 某些流行病传播模型的对偶性

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.2025.104773

C. Franceschini , E. Saada , G.M. Schütz , S. Velasco

We examine the role of boundaries and the structure of nontrivial duality functions for three non-conservative interacting particle systems in one dimension that model epidemic spreading: (i) the diffusive contact process (DCP), (ii) a model that we introduce and call generalized diffusive contact process (GDCP), both in finite volume in contact with boundary reservoirs, i.e., with open boundaries, and (iii) the susceptible–infectious–recovered (SIR) model on

Z

. We establish duality relations for each system through an analytical approach. It turns out that with open boundaries self-duality breaks down and qualitatively different properties compared to closed boundaries (i.e., finite volume without reservoirs) arise: Both the DCP and GDCP are ergodic but no longer absorbing, while the respective dual processes are absorbing but not ergodic. We provide expressions for the stationary correlation functions in terms of the dual absorption probabilities. We perform explicit computations for a small sized DCP, and for arbitrary size in a particular setting of the GDCP. The duality function is factorized for the DCP and GDCP, contrary to the SIR model for which the duality relation is nonlocal and yields an explicit expression of the time evolution of some specific correlation functions, describing the time decay of the sizes of clusters of susceptible individuals.

我们研究了三维非保守相互作用粒子系统中边界的作用和非平凡对偶函数的结构，这些系统可以模拟流行病的传播：(i)扩散接触过程（DCP），（ii）我们引入并称为广义扩散接触过程（GDCP）的模型，两者都是在有限体积内与边界水库接触，即开放边界，以及（iii） z上的易感-感染-恢复（SIR）模型。我们通过分析方法建立了每个系统的对偶关系。结果表明，在开放边界下，自对偶性被打破，与封闭边界（即没有储层的有限体积）相比，出现了质的不同：DCP和GDCP都是遍历的，但不再吸收，而各自的对偶过程是吸收的，但不是遍历的。我们用对偶吸收概率给出了平稳相关函数的表达式。我们对小尺寸的DCP进行了显式计算，并对GDCP的特定设置中的任意尺寸进行了显式计算。对DCP和GDCP的对偶函数进行因式分解，与SIR模型相反，SIR模型的对偶关系是非局部的，并且产生了一些特定相关函数的时间演化的显式表达式，描述了易感个体簇大小的时间衰减。

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引用次数: 0

Deviation inequalities for contractive infinite memory processes 收缩无限存储过程的偏差不等式

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-25 DOI: 10.1016/j.spa.2025.104778

Paul Doukhan , Xiequan Fan

In this paper, we introduce a class of stochastic processes that encompasses many natural and widely used examples. A key feature of these processes is their infinite memory, which enables them to retain information from arbitrarily distant past states. Using the martingale decomposition method, we derive deviation and moment inequalities for separately Lipschitz functionals of such processes, under various moment conditions on certain dominating random variables. Our results extend those obtained for Markov chains by Dedecker and Fan [Stochastic Process. Appl., 2015], as well as recent results by Chazottes et al. [Ann. Appl. Probab., 2023] concerning specific infinite-memory models with sub-Gaussian concentration bounds. We also discuss an application to the stochastic gradient Langevin dynamics algorithm.

在本文中，我们介绍了一类随机过程，它包含了许多自然的和广泛使用的例子。这些过程的一个关键特征是它们的无限记忆，这使它们能够从任意遥远的过去状态中保留信息。利用鞅分解方法，我们分别导出了这类过程的Lipschitz泛函在不同的矩条件下对某些主导随机变量的偏差不等式和矩不等式。我们的结果推广了用Dedecker和Fan[随机过程]得到的关于马尔可夫链的结果。达成。， 2015]，以及Chazottes等人最近的研究结果。达成。Probab。[j]，[2023]关于具有亚高斯浓度边界的特定无限记忆模型。我们还讨论了随机梯度朗之万动力学算法的一个应用。

引用次数: 0

Holomorphic jump-diffusions 全纯jump-diffusions

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-24 DOI: 10.1016/j.spa.2025.104781

Christa Cuchiero , Francesca Primavera , Sara Svaluto-Ferro

We introduce a class of jump-diffusions, called holomorphic, of which the well-known classes of affine and polynomial processes are particular instances. The defining property concerns the extended generator, which is required to map a (subset of) holomorphic functions to themselves. This leads to a representation of the expectation of power series of the process’ marginals via a potentially infinite dimensional linear ODE. We apply the same procedure by considering exponentials of holomorphic functions, leading to a class of processes named affine-holomorphic for which a representation for quantities as the characteristic function of power series is provided. Relying on powerful results from complex analysis, we obtain sufficient conditions on the process’ characteristics which guarantee the holomorphic and affine-holomorphic properties and provide applications to several classes of jump-diffusions.

我们引入了一类跳跃扩散，称为全纯，其中众所周知的仿射和多项式过程是它的特殊实例。定义属性涉及扩展生成器，它需要将全纯函数（子集）映射到自身。这导致通过潜在的无限维线性ODE表示过程边际的幂级数的期望。我们用同样的方法考虑全纯函数的指数，得到了一类被称为仿射全纯的过程，给出了量作为幂级数特征函数的表示。依靠复分析的有力结果，我们得到了保证全纯和仿射全纯性质的过程特征的充分条件，并为几类跳跃扩散提供了应用。

引用次数: 0

Swarm dynamics for global optimization on finite sets 有限集上全局优化的群体动力学

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-22 DOI: 10.1016/j.spa.2025.104780

Nhat-Thang Le , Laurent Miclo

Consider the global optimisation of a function

U

defined on a finite set

V

endowed with an irreducible and reversible Markov generator. By integration, we extend

U

to the set

P (V)

of probability distributions on

V

and we penalize it with a time-dependent generalized entropy functional. Endowing

P (V)

with a Maas’ Wasserstein-type Riemannian structure enables us to consider an associated time-inhomogeneous gradient descent algorithm. There are several ways to interpret this

P (V)

-valued dynamical system as the time-marginal laws of a time-inhomogeneous non-linear Markov process taking values in

V

, each of them allowing for interacting particle approximations. This procedure extends to the discrete framework the continuous state space swarm algorithm approach of Bolte et al. (2023), but here we go further by considering more general generalized entropy functionals for which functional inequalities can be proven. Thus in the full generality of the above finite framework, we give conditions on the underlying time dependence ensuring the convergence of the algorithm toward laws supported by the set of global minima of

U

. Numerical simulations illustrate that one has to be careful about the choice of the time-inhomogeneous non-linear Markov process interpretation.

考虑函数U在有限集合V上的全局优化问题，该集合具有不可约可逆马尔可夫生成器。通过积分，我们将U扩展到V上概率分布的集合P(V)，并用一个与时间相关的广义熵泛函来惩罚它。赋予P(V)一个Maas ' wasserstein型黎曼结构使我们能够考虑一个相关的时间非齐次梯度下降算法。有几种方法可以将这个P(V)值的动力系统解释为时间非齐次非线性马尔可夫过程的时间边缘定律，其值为V，每种方法都允许相互作用的粒子近似。该过程将Bolte等人（2023）的连续状态空间群算法方法扩展到离散框架，但这里我们进一步考虑可以证明函数不等式的更一般的广义熵函数。因此，在上述有限框架的充分普遍性下，我们给出了基本时间依赖性的条件，以确保算法收敛于u的全局最小值集所支持的定律。数值模拟表明，人们必须小心选择时间非齐次非线性马尔可夫过程解释。

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引用次数: 0

Optimal control of the nonlinear stochastic Fokker–Planck equation 非线性随机Fokker-Planck方程的最优控制

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-17 DOI: 10.1016/j.spa.2025.104774

Ben Hambly , Philipp Jettkant

We consider a control problem for the nonlinear stochastic Fokker–Planck equation. This equation describes the evolution of the distribution of nonlocally interacting particles affected by a common source of noise. The system is directed by a controller that acts on the drift term with the goal of minimising a cost functional. We establish the well-posedness of the state equation, prove the existence of optimal controls, and formulate a stochastic maximum principle (SMP) that provides necessary and sufficient optimality conditions for the control problem. The adjoint process arising in the SMP is characterised by a nonlocal (semi)linear backward SPDE for which we study existence and uniqueness. We also rigorously connect the control problem for the nonlinear stochastic Fokker–Planck equation to the control of the corresponding McKean–Vlasov SDE that describes the motion of a representative particle. Our work extends existing results for the control of the Fokker–Planck equation to nonlinear and stochastic dynamics. In particular, the sufficient SMP, which we obtain by exploiting the special structure of the Fokker–Planck equation, seems to be novel even in the linear deterministic setting. We illustrate our results with an application to a model of government interventions in financial systems, supplemented by numerical illustrations.

考虑非线性随机Fokker-Planck方程的控制问题。这个方程描述了受共同噪声源影响的非局部相互作用粒子的分布演化。系统由控制器控制，控制器作用于漂移项，目标是最小化成本函数。我们建立了状态方程的适定性，证明了最优控制的存在性，并构造了一个随机极大值原理（SMP），为控制问题的最优性提供了充分必要条件。SMP中出现的伴随过程具有非局部（半）线性后向SPDE的特征，我们研究了其存在唯一性。我们还将非线性随机Fokker-Planck方程的控制问题严格地与描述代表性粒子运动的相应McKean-Vlasov SDE的控制联系起来。我们的工作将现有的控制福克-普朗克方程的结果扩展到非线性和随机动力学。特别是，我们利用Fokker-Planck方程的特殊结构得到的充分SMP，即使在线性确定性设置下也显得新颖。我们通过应用政府干预金融系统的模型来说明我们的结果，并辅以数字插图。

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引用次数: 0

Convergence of adapted smoothed empirical measures 自适应平滑经验测度的收敛性

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-17 DOI: 10.1016/j.spa.2025.104775

Songyan Hou

The adapted Wasserstein distance (

AW

-distance) controls the calibration errors of optimal values in various stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. However, statistical aspects of the

AW

-distance are bottlenecked by the failure of empirical measures (Emp) to converge under this distance. Kernel smoothing and adapted projection have been introduced to construct converging substitutes of empirical measures, known respectively as smoothed empirical measures (

S

-Emp) and adapted empirical measures (

A

-Emp). However, both approaches have limitations. Specifically,

S

-Emp lack comprehensive convergence results, whereas

A

-Emp in practical applications lead to fewer distinct samples compared to standard empirical measures.

In this work, we address both of the aforementioned issues. First, we develop comprehensive convergence results of

S

-Emp. We then introduce a smoothed version for

A

-Emp, which provide as many distinct samples as desired. We refer them as

AS

-Emp and establish their convergence in mean, deviation and almost sure convergence. The convergence estimation incorporates two results: the empirical analysis of the smoothed adapted Wasserstein distance (

{AW}^{(σ)}

-distance) and its bandwidth effects. Both results are novel and their proof techniques could be of independent interest.

自适应Wasserstein距离（AW-distance）控制了各种随机优化问题、定价与套期保值问题、最优停止问题等最优值的校准误差。然而，由于经验测度（Emp）不能在此距离下收敛，使得aw距离的统计方面受到瓶颈。引入核平滑和自适应投影来构造经验测度的收敛替代品，分别称为平滑经验测度（S-Emp）和自适应经验测度（A-Emp）。然而，这两种方法都有局限性。具体而言，S-Emp缺乏全面的收敛结果，而A-Emp在实际应用中与标准经验测量相比，不同的样本较少。在这项工作中，我们解决了上述两个问题。首先，我们得到了S-Emp的综合收敛结果。然后，我们介绍了a - emp的平滑版本，它提供了尽可能多的不同样本。我们将其称为as - emp，并建立了它们在均值、偏差和几乎肯定收敛方面的收敛性。收敛性估计包含两个结果：平滑自适应Wasserstein距离（AW(σ)-距离）及其带宽效应的实证分析。这两个结果都是新颖的，它们的证明技术可能具有独立的兴趣。

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引用次数: 0

Mixing time and cutoff for the k-SPEP k-SPEP的混合时间和截止时间

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-16 DOI: 10.1016/j.spa.2025.104776

Eyob Tsegaye

We investigate the mixing time of the capacity

k

symmetric partial exclusion process of Schütz and Sandow with

m

particles on a segment of length

N

, and we show that this process exhibits cutoff at time

\frac{1}{2 k π^{2}} N^{2} log m

. We also introduce a related complete multi-species process that we call the

S_{k, N}

shuffle and show that this process exhibits cutoff at time

\frac{1}{2 k π^{2}} N^{2} log (N)

. This extends the celebrated result of Lacoin, which proved cutoff for the symmetric simple exclusion process on a segment of length

N

and the adjacent transposition shuffle.

研究了sch tz和Sandow在长度为N的段上与m粒子的容量k对称部分不相容过程的混合时间，结果表明该过程在12kπ2N2logm时出现截断。我们还介绍了一个相关的完整的多物种过程，我们称之为Sk，N洗牌，并表明该过程在时间12kπ2N2log(N)时呈现截止。这扩展了Lacoin的著名结果，该结果证明了对称简单不相容过程在长度为N的段和相邻的转置洗牌上的截止性。

引用次数: 0

Reflected BSDE driven by a marked point process with a convex/concave generator 反射BSDE驱动的标记点过程与凸/凹生成器

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-16 DOI: 10.1016/j.spa.2025.104777

Zihao Gu , Yiqing Lin , Kun Xu

In this paper, we study a class of reflected backward stochastic differential equations (RBSDE) driven by a marked point process (MPP) with a convex/concave generator. Based on fixed point argument,

θ

-method and truncation technique, the well-posedness of this kind of RBSDE with unbounded terminal condition and obstacle is investigated. Besides, we present an application on the pricing of American options via utility maximization, which is solved by constructing an RBSDE with a convex generator.

研究了一类由凸/凹发生器标记点过程驱动的反射后向随机微分方程（RBSDE）。利用不动点论证、θ-法和截断技术，研究了这类具有无界终端条件和障碍的RBSDE的适定性。此外，我们还构造了一个带凸生成器的RBSDE来求解效用最大化的美式期权定价问题。

引用次数: 0

A tamed Euler scheme for SDEs with non-locally integrable drift coefficient 漂移系数非局部可积的SDEs的驯服欧拉格式

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-16 DOI: 10.1016/j.spa.2025.104772

Tim Johnston , Sotirios Sabanis

In this article we show that for SDEs with a drift coefficient that is non-locally integrable, one may define a tamed Euler scheme that converges in

L^{p}

at rate

1 / 2

to the true solution. The taming is required in this case since one cannot expect the regular Euler scheme to have finite moments in

L^{p}

. Our proof strategy involves controlling the inverse moments of the distance of scheme and the true solution to the singularity set. We additionally show that our setting applies to the case of two scalar valued particles with singular interaction kernel. To the best of the authors’ knowledge, this is the first work to prove strong convergence of an Euler-type scheme in the case of non-locally integrable drift.

在本文中，我们证明了对于具有非局部可积漂移系数的SDEs，可以定义一个在Lp中以1/2的速率收敛到真解的驯化欧拉格式。在这种情况下，由于不能期望正则欧拉格式在Lp中具有有限矩，因此需要进行驯服。我们的证明策略包括控制方案距离的逆矩和奇异集的真解。此外，我们还证明了我们的设置适用于具有奇异相互作用核的两个标量值粒子的情况。据作者所知，这是第一个证明欧拉型格式在非局部可积漂移情况下的强收敛性的工作。

引用次数: 0

Spectral gap for the stochastic exchange model 随机交换模型的谱隙

IF 1.2 2区数学 Q3 STATISTICS & PROBABILITY

Stochastic Processes and their Applications

Pub Date : 2025-09-12 DOI: 10.1016/j.spa.2025.104769

Eric A. Carlen , Gustavo Posta , Imre Péter Tóth

We prove a spectral gap inequality for the stochastic exchange model studied by Gaspard and Gilbert and by Grigo, Khanin and Szász in connection with understanding heat conduction in a deterministic billiards model. The bound on the spectral gap that we prove is uniform in the number of particles, as had been conjectured. We adapt techniques that were originally developed to prove spectral gap bounds for the Kac model with hard sphere collisions, which, like the stochastic exchange model, has degenerate jump rates.

我们证明了Gaspard和Gilbert以及Grigo， Khanin和Szász研究的随机交换模型的谱间隙不等式，并与理解确定性台球模型中的热传导有关。我们所证明的谱隙的界在粒子数上是一致的，正如我们所推测的那样。我们采用了最初开发的技术来证明具有硬球碰撞的Kac模型的谱间隙界限，硬球碰撞与随机交换模型一样，具有退化跳变率。

引用次数: 0

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