58 research outputs found
Two-sided Gaussian bounds for fundamental solutions of non-divergence form parabolic operators with H"older continuous coefficients
No full text
Non-Arbitrage under a Class of Honest Times
No full textThis paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping with the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature —up to our knowledge— proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore , we construct explicitly " the-after-τ " local martingale deflators for a large class of initial models (i.e.,models in the small filtration) that are already risk-neutralized
Двойная логарифмическая устойчивость в идентификации скалярного потенциала по частичной эллиптической карте Дирихле - Неймана
No full textМурад Чулли, профессор, Университет Лотарингии (г. Мец, Франция),
[email protected].
Явар Киан, профессор, Университет Экс-Марсель, (г. Марсель, Франция),
[email protected].
Эрик Соккорси, профессор, Университет Экс-Марсель, (г. Марсель, Франция),
[email protected]. M. Choulli, University of Lorraine, Metz, France,
[email protected],
Y. Kian, Aix-Marseille University, Marseille, France, [email protected],
E. Soccorsi, Aix-Marseille University, Marseille, France, [email protected]Исследуется вопрос устойчивости решения обратной задачи определения скалярного потенциала, возникающего в стационарном уравнении Шредингера в ограниченной области по частичной эллиптической карте Дирихле - Неймана. А именно, условия Дирихле ставятся на затененной части границы области и условия Неймана - на ее освещенной части. Установлена оценка устойчивости типа loglog для L2-нормы (соотв. H^1 нормы) для H*, при t > 0 и ограниченых (соотв. L2) потенциалов. We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr odinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated
face. We establish a log log stability estimate for the L2-norm (resp. the H1-norm) of Ht, for t > 0, and bounded (resp. L2) potentials
Stability estimate for a semilinear elliptic inverse problem
No full textWe establish a logarithmic stability estimate for the inverse problem of determining the nonlinear term, appearing in a semilinear boundary value problem, from the corresponding Dirichlet-to-Neumann map. Our result can be seen as a stability inequality for an earlier uniqueness result by Isakov and Sylvester (Commun Pure Appl Math 47:1403–1410, 1994). © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG
New global logarithmic stability results on the Cauchy problem for elliptic equations
No full textI
DOI:
10.1017/S000497271900078
Boundary value problems for elliptic partial differential equations
No full textMasterThis course is intended as an introduction to the analysis of elliptic partial differential equations. The objective is to provide a large overview of the different aspects of elliptic partial differential equations and their modern treatment. Besides variational and Schauder methods we study the unique continuation property and the stability for Cauchy problems. The derivation of the unique continuation property and the stability for Cauchy problems relies on a Carleman inequality. This inequality is efficient to establish three-ball type inequalities which are the main tool in the continuation argument. We know that historically a central role in the analysis of partial differential equations is played by their fundamental solutions. We added an appendix dealing with the construction of a fundamental solution by the so-called Levi parametrix method. We tried as much as possible to render this course self-contained. Moreover each chapter contains many exercices and problems. We have provided detailed solutions of these exercises and problems.The most parts of this course consist in an enhanced version of courses given by the author in both undergraduate and graduate levels during several years. Remarks and comments that can help to improve this course are welcome
Abstract inverse problem and application
No full textAbstractIn this paper we consider the determination of the unknown function ƒ: [0, T] → X (X is a Banach space), in the abstract Cauchy problem u′(t) = Au(t) + ƒ(t) and u(0) = x, from an additional given data. First, we establish an existence and uniqueness theorem when ƒ(·) = ∑i = 1n hi(·) yi, where hi, 1 ⩽ i ⩽ n, are unknown numerical functions and yi ϵ X, 1 ⩽ i ⩽ n, are known. This theorem is then used to derive a nonuniqueness result for the class of functions which are not written in the form ƒ(·) = ∑i=1m ki(·)zi, ki: [0, T] → C and zi ϵ X. We also show the continuous dependence of the inverse problem solution on the additional data. Finally, an application to the parabolic equation is indicated
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