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Initial-boundary value problem for transport equations driven by rough paths 粗糙路径驱动的传输方程的初始边界值问题
IF 0.9 Q3 Mathematics Pub Date : 2024-05-10 DOI: 10.1090/tpms/1212
Dai Noboriguchi
In this paper, we are interested in the initial Dirichlet boundary value problem for a transport equation driven by weak geometric Hölder p p -rough paths. We introduce a notion of solutions to rough partial differential equations with boundary conditions. Consequently, we will establish a well-posedness for such a solution under some assumptions stated below. Moreover, the solution is given explicitly.
在本文中,我们关注的是由弱几何荷尔德 p p - 通过路径驱动的传输方程的初始 Dirichlet 边界值问题。我们引入了带边界条件的粗糙偏微分方程解的概念。因此,我们将在下文所述的一些假设条件下建立这样一个解的好求解性。此外,我们还将明确给出解。
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引用次数: 0
Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients 带算子系数的二阶差分方程的均值有界和静态解
IF 0.9 Q3 Mathematics Pub Date : 2024-05-10 DOI: 10.1090/tpms/1211
M. Horodnii
We study the question of the existence of a unique bounded in the mean solution for the second-order difference equation with piecewise constant operator coefficients and of the stationary solution of the corresponding difference equation with constant operator coefficients. The case is considered when the corresponding “algebraic” operator equations have separated roots.
我们研究了具有片断常数算子系数的二阶差分方程的唯一有界均值解的存在性问题,以及具有常数算子系数的相应差分方程的静止解的存在性问题。当相应的 "代数 "算子方程具有分离的根时,我们将考虑这种情况。
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引用次数: 0
The Burgers-type equation driven by a stochastic measure 由随机测量驱动的布尔格斯型方程
IF 0.9 Q3 Mathematics Pub Date : 2024-05-10 DOI: 10.1090/tpms/1213
Vadym Radchenko
We study the one-dimensional equation driven by a stochastic measure μ mu . For μ mu we assume only σ sigma -additivity in probability. Our results imply the global existence and uniqueness of the solution to the heat equation and the local existence and uniqueness of the solution to the Burgers equation. The averaging principle for such equation is studied.
我们研究由随机度量 μ mu 驱动的一元方程。对于 μ mu,我们只假设概率的 σ σ -加性。我们的结果意味着热方程解的全局存在性和唯一性,以及布尔格斯方程解的局部存在性和唯一性。研究了此类方程的平均原理。
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引用次数: 0
Non-adaptive estimation for degenerate diffusion processes 退化扩散过程的非适应性估计
IF 0.9 Q3 Mathematics Pub Date : 2024-05-10 DOI: 10.1090/tpms/1207
A. Gloter, Nakahiro Yoshida
We consider a degenerate system of stochastic differential equations. The first component of the system has a parameter θ 1 theta _1 in a non-degenerate diffusion coefficient and a parameter θ 2 theta _2 in the drift term. The second component has a drift term with a parameter θ 3 theta _3 and no diffusion term. Parametric estimation of the degenerate diffusion system is discussed under a sampling scheme. We investigate the asymptotic behavior of the joint quasi-maximum likelihood estimator for ( θ 1 , θ 2 , θ 3 ) (theta _1,theta _2,theta _3) . The estimation scheme is non-adaptive. The estimator incorporates information of the increments of both components, and under this construction, we show that the asymptotic variance of the estimator for θ 1 theta _1 is smaller than the one for standard estimator based on the first component only, and that the convergence of the estimator for θ 3 theta _3 is much faster than for the other parameters. By simulation studies, we compare the performance of the joint quasi-maximum likelihood estimator with the adaptive and one-step estimators investigated in Gloter and Yoshida [Electron. J. Statist 15 (2021), no. 1, 1424–1472].
我们考虑一个退化的随机微分方程系统。系统的第一个分量的非退化扩散系数有一个参数 θ 1 theta _1,漂移项有一个参数 θ 2 theta _2。第二个分量的漂移项参数为 θ 3 theta _3,没有扩散项。在采样方案下讨论了退化扩散系统的参数估计。我们研究了 ( θ 1 , θ 2 , θ 3 ) (theta _1,theta _2,theta _3) 的联合准极大似然估计器的渐近行为。估计方案是非适应性的。在这种结构下,我们发现θ 1 theta _1的估计值的渐近方差小于仅基于第一个分量的标准估计值的渐近方差,而且θ 3 theta _3的估计值的收敛速度远快于其他参数。通过模拟研究,我们比较了联合准极大似然估计器与 Gloter 和 Yoshida [Electron. J. Statist 15 (2021),no. 1,1424-1472] 中研究的自适应估计器和一步估计器的性能。
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引用次数: 0
Characterization of the least squares estimator: Mis-specified multivariate isotonic regression model with dependent errors 最小二乘估计器的特征:带有因果误差的多变量同调回归模型的错误定义
IF 0.9 Q3 Mathematics Pub Date : 2024-05-10 DOI: 10.1090/tpms/1210
Pramita Bagchi, Subhra Dhar
This article investigates some nice properties of the least squares estimator of multivariate isotonic regression function (denoted as LSEMIR), when the model is mis-specified, and the errors are β beta -mixing stationary random variables. Under mild conditions, it is observed that the least squares estimator converges uniformly to a certain monotone function, which is closest to the original function in an appropriate sense.
本文研究了多元等调回归函数(记为 LSEMIR)的最小二乘估计器在模型被错误指定、误差为 β beta 混合静态随机变量时的一些良好性质。在温和的条件下,可以观察到最小二乘估计器均匀地收敛于某个单调函数,在适当的意义上最接近原始函数。
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引用次数: 0
Temporal properties of the stochastic fractional heat equation with spatially-colored noise 带有空间色噪声的随机分数热方程的时间特性
IF 0.9 Q3 Mathematics Pub Date : 2024-05-10 DOI: 10.1090/tpms/1209
Ran Wang, Yimin Xiao

Consider the stochastic partial differential equation t u t ( x ) = ( Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ (

考虑随机偏微分方程 ∂ t u t ( x ) = - ( - Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ ( t , x ) , t ≥ 0 , x∈ R d , begin{equation*}frac {partial }{partial t}u_t(boldsymbol {x})= -(-Delta )^{frac {alpha }{2}}u_t(boldsymbol {x}) +bleft (u_t(boldsymbol {x})right ) +sigma left (u_t(boldsymbol {x})right )dot F(t, boldsymbol {x}), quad tge 0,:boldsymbol {x}in mathbb R^d, end{equation*} 其中 - ( - Δ ) α 2 -(-Delta )^{frac{alpha}{2}}表示分数拉普拉斯幂 α 2∈ ( 1 2 , 1 ]。 frac {alpha }{2}in (frac 12,1] , 和驱动噪声 F ˙frac {alpha }{2}in (frac 12,1]. 驱动噪声 F ˙ dot F 是一个居中的高斯场,在时间上是白色的,在空间上具有由 Riesz 核给出的同质协方差。我们研究在任意固定的 t > 0 t > 0 且 x∈ R d boldsymbol {x}in mathbb R^d 时,时间梯度 u t + ε ( x ) - u t ( x ) u_{t+{varepsilon }}(boldsymbol{x})-u_t(boldsymbol{x})近似的详细行为,当 ε ↓ 0 {varepsilon }downarrow 0 时。作为应用,我们推导出 Khintchin 的迭代对数定律、Chung 的迭代对数定律,以及关于时间过程 { u t ( x ) 的 q q - 变量的结果。
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引用次数: 0
Full inference for the anisotropic fractional Brownian field 各向异性分数布朗场的完全推论
IF 0.9 Q3 Mathematics Pub Date : 2024-05-10 DOI: 10.1090/tpms/1204
Paul Escande, Frédéric Richard
The anisotropic fractional Brownian field (AFBF) is a non-stationary Gaussian random field which has been used for the modeling of textured images. In this paper, we address the open issue of estimating the functional parameters of this field, namely the topothesy and Hurst functions. We propose an original method which fits the empirical semi-variogram of an image to the semi-variogram of a turning-band field that approximates the AFBF. Expressing the fitting criterion in terms of a separable non-linear least square criterion, we design a minimization algorithm inspired by the variable projection approach. This algorithm also includes a coarse-to-fine multigrid strategy based on approximations of functional parameters. Compared to existing methods, the new method enables to estimate both functional parameters on their whole definition domain. On simulated textures, we show that it has a low estimation error, even when the parameters are approximated with a high precision. We also apply the method to characterize mammograms and sample images with synthetic parenchymal patterns.
各向异性分数布朗场(AFBF)是一种非稳态高斯随机场,已被用于纹理图像建模。在本文中,我们将探讨如何估算该场的函数参数,即拓扑函数和赫斯特函数。我们提出了一种独创的方法,将图像的经验半变量图与近似于 AFBF 的转带场的半变量图进行拟合。我们用可分离的非线性最小平方准则来表示拟合准则,设计了一种受变量投影法启发的最小化算法。该算法还包括一种基于函数参数近似值的从粗到细多网格策略。与现有方法相比,新方法能够在整个定义域内估算两个函数参数。在模拟纹理上,我们发现即使参数近似精度很高,该方法的估计误差也很低。我们还将该方法应用于描述乳房 X 线照片和具有合成实质模式的样本图像。
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引用次数: 0
Stochastic differential equations with discontinuous diffusion coefficients 具有不连续扩散系数的随机微分方程
Q3 Mathematics Pub Date : 2023-10-03 DOI: 10.1090/tpms/1201
Soledad Torres, Lauri Viitasaari
We study one-dimensional stochastic differential equations of the form d X t = σ ( X t ) d Y t dX_t = sigma (X_t)dY_t , where Y Y is a suitable Hölder continuous driver such as the fractional Brownian motion B H B^H with H > 1 2 H>frac 12 . The innovative aspect of the present paper lies in the assumptions on diffusion coefficients σ sigma for which we assume very mild conditions. In particular, we allow σ sigma to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.
我们研究了dX t = σ (X t)dY t dX_t = sigma (X_t)dY_t的一维随机微分方程,其中Y Y是一个合适的Hölder连续驱动器,如分数阶布朗运动B H B^H with H &gt;12 H&gt;frac本文的创新之处在于对扩散系数σ sigma的假设,我们假设了非常温和的条件。特别地,我们允许σ sigma具有不连续,因此我们的结果可以应用于研究具有不连续扩散的方程。
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引用次数: 0
Reverse stress testing in skew-elliptical models 斜椭圆模型的反向应力测试
Q3 Mathematics Pub Date : 2023-10-03 DOI: 10.1090/tpms/1199
Jonathan von Schroeder, Thorsten Dickhaus, Taras Bodnar
Stylized facts about financial data comprise skewed and heavy-tailed (log-)returns. Therefore, we revisit previous results on reverse stress testing under elliptical models, and we extend them to the broader class of skew-elliptical models. In the elliptical case, an explicit formula for the solution is provided. In the skew-elliptical case, we characterize the solution in terms of an easy-to-implement numerical optimization problem. As specific examples, we investigate the classes of skew-normal and skew-t models in detail. Since the solutions depend on population parameters, which are often unknown in practice, we also tackle the statistical task of estimating these parameters and provide confidence regions for the most likely scenarios.
关于金融数据的程式化事实包括倾斜和重尾(log-)回报。因此,我们回顾了以前在椭圆模型下的反向应力测试结果,并将它们扩展到更广泛的斜椭圆模型类。在椭圆情况下,给出了解的显式公式。在斜椭圆的情况下,我们用一个易于实现的数值优化问题来描述解。作为具体的例子,我们详细研究了斜正态和斜t模型的类别。由于解依赖于总体参数,而这些参数在实践中通常是未知的,因此我们还处理估计这些参数的统计任务,并为最可能的场景提供置信区域。
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引用次数: 0
Distributional hyperspace-convergence of Argmin-sets in convex 𝑀-estimation 凸上argmin集的分布超空间收敛性𝑀-estimation
Q3 Mathematics Pub Date : 2023-10-03 DOI: 10.1090/tpms/1195
Dietmar Ferger
In M M -estimation we consider the sets of all minimizing points of convex empirical criterion functions. These sets are random closed sets. We derive distributional convergence in the hyperspace of all closed subsets of the real line endowed with the Fell-topology. As a special case single minimizing points converge in distribution in the classical sense. In contrast to the literature so far, unusual rates of convergence and non-normal limits emerge, which go far beyond the square-root asymptotic normality. Moreover, our theory can be applied to the sets of zero-estimators.
在M -估计中,我们考虑凸经验准则函数的所有极小点的集合。这些集合是随机闭集。我们得到了具有fell拓扑的实线的所有闭子集在超空间中的分布收敛性。作为一种特殊情况,单极小点在经典意义上的分布是收敛的。与迄今为止的文献相反,出现了不寻常的收敛速度和非正态极限,它们远远超出了平方根渐近正态。此外,我们的理论可以应用于零估计量的集合。
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引用次数: 2
期刊
Theory of Probability and Mathematical Statistics
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