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。
{"title":"Initial-boundary value problem for transport equations driven by rough paths","authors":"Dai Noboriguchi","doi":"10.1090/tpms/1212","DOIUrl":"https://doi.org/10.1090/tpms/1212","url":null,"abstract":"In this paper, we are interested in the initial Dirichlet boundary value problem for a transport equation driven by weak geometric Hölder \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140993862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients","authors":"M. Horodnii","doi":"10.1090/tpms/1211","DOIUrl":"https://doi.org/10.1090/tpms/1211","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140990790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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。
{"title":"The Burgers-type equation driven by a stochastic measure","authors":"Vadym Radchenko","doi":"10.1090/tpms/1213","DOIUrl":"https://doi.org/10.1090/tpms/1213","url":null,"abstract":"We study the one-dimensional equation driven by a stochastic measure \u0000\u0000 \u0000 μ\u0000 mu\u0000 \u0000\u0000. For \u0000\u0000 \u0000 μ\u0000 mu\u0000 \u0000\u0000 we assume only \u0000\u0000 \u0000 σ\u0000 sigma\u0000 \u0000\u0000-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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140992627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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].
{"title":"Non-adaptive estimation for degenerate diffusion processes","authors":"A. Gloter, Nakahiro Yoshida","doi":"10.1090/tpms/1207","DOIUrl":"https://doi.org/10.1090/tpms/1207","url":null,"abstract":"We consider a degenerate system of stochastic differential equations. The first component of the system has a parameter \u0000\u0000 \u0000 \u0000 θ\u0000 1\u0000 \u0000 theta _1\u0000 \u0000\u0000 in a non-degenerate diffusion coefficient and a parameter \u0000\u0000 \u0000 \u0000 θ\u0000 2\u0000 \u0000 theta _2\u0000 \u0000\u0000 in the drift term. The second component has a drift term with a parameter \u0000\u0000 \u0000 \u0000 θ\u0000 3\u0000 \u0000 theta _3\u0000 \u0000\u0000 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 \u0000\u0000 \u0000 \u0000 (\u0000 \u0000 θ\u0000 1\u0000 \u0000 ,\u0000 \u0000 θ\u0000 2\u0000 \u0000 ,\u0000 \u0000 θ\u0000 3\u0000 \u0000 )\u0000 \u0000 (theta _1,theta _2,theta _3)\u0000 \u0000\u0000. 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 \u0000\u0000 \u0000 \u0000 θ\u0000 1\u0000 \u0000 theta _1\u0000 \u0000\u0000 is smaller than the one for standard estimator based on the first component only, and that the convergence of the estimator for \u0000\u0000 \u0000 \u0000 θ\u0000 3\u0000 \u0000 theta _3\u0000 \u0000\u0000 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].","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140993463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Characterization of the least squares estimator: Mis-specified multivariate isotonic regression model with dependent errors","authors":"Pramita Bagchi, Subhra Dhar","doi":"10.1090/tpms/1210","DOIUrl":"https://doi.org/10.1090/tpms/1210","url":null,"abstract":"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 \u0000\u0000 \u0000 β\u0000 beta\u0000 \u0000\u0000-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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140991011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the stochastic partial differential equation ∂∂tut(x)=−(−Δ)α2ut(x)+b(ut(x))+σ(ut(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。
{"title":"Temporal properties of the stochastic fractional heat equation with spatially-colored noise","authors":"Ran Wang, Yimin Xiao","doi":"10.1090/tpms/1209","DOIUrl":"https://doi.org/10.1090/tpms/1209","url":null,"abstract":"<p>Consider the stochastic partial differential equation <disp-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartFraction partial-differential Over partial-differential t EndFraction u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis equals minus left-parenthesis negative normal upper Delta right-parenthesis Superscript StartFraction alpha Over 2 EndFraction Baseline u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis plus b left-parenthesis u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis right-parenthesis plus sigma left-parenthesis u Subscript t Baseline left-parenthesis bold-italic x right-parenthesis right-parenthesis ModifyingAbove upper F With dot left-parenthesis t comma bold-italic x right-parenthesis comma t greater-than-or-equal-to 0 comma bold-italic x element-of double-struck upper R Superscript d Baseline comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mfrac>\u0000 <mml:mi mathvariant=\"normal\">∂</mml:mi>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∂</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mfrac>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo>−</mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mo>−</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Δ</mml:mi>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mfrac>\u0000 <mml:mi>α</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mfrac>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>b</mml:mi>\u0000 <mml:mrow>\u0000 <mml:mo>(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>σ</mml:mi>\u0000 <mml:mrow>\u0000 <mml:mo>(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi mathvariant=\"bold-italic\">x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:mo>˙</mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140992797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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。
{"title":"Full inference for the anisotropic fractional Brownian field","authors":"Paul Escande, Frédéric Richard","doi":"10.1090/tpms/1204","DOIUrl":"https://doi.org/10.1090/tpms/1204","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140992207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study one-dimensional stochastic differential equations of the form dXt=σ(Xt)dYtdX_t = sigma (X_t)dY_t, where YY is a suitable Hölder continuous driver such as the fractional Brownian motion BHB^H with H>12H>frac 12. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients σsigma for which we assume very mild conditions. In particular, we allow σsigma 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。
{"title":"Stochastic differential equations with discontinuous diffusion coefficients","authors":"Soledad Torres, Lauri Viitasaari","doi":"10.1090/tpms/1201","DOIUrl":"https://doi.org/10.1090/tpms/1201","url":null,"abstract":"We study one-dimensional stochastic differential equations of the form <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d upper X Subscript t Baseline equals sigma left-parenthesis upper X Subscript t Baseline right-parenthesis d upper Y Subscript t\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dX_t = sigma (X_t)dY_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y\"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=\"application/x-tex\">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a suitable Hölder continuous driver such as the fractional Brownian motion <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Superscript upper H\"> <mml:semantics> <mml:msup> <mml:mi>B</mml:mi> <mml:mi>H</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">B^H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H greater-than one half\"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">H>frac 12</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which we assume very mild conditions. In particular, we allow <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Reverse stress testing in skew-elliptical models","authors":"Jonathan von Schroeder, Thorsten Dickhaus, Taras Bodnar","doi":"10.1090/tpms/1199","DOIUrl":"https://doi.org/10.1090/tpms/1199","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In MM-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.
{"title":"Distributional hyperspace-convergence of Argmin-sets in convex 𝑀-estimation","authors":"Dietmar Ferger","doi":"10.1090/tpms/1195","DOIUrl":"https://doi.org/10.1090/tpms/1195","url":null,"abstract":"In <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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