Theory of Probability and Mathematical Statistics最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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>&gt;</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&gt;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}

{"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}

{"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}