209 results on '"46A20"'
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2. On the Grothendieck duality for the space of holomorphic Sobolev functions
- Author
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Levskii, Arkadii and Shlapunov, Alexander
- Subjects
Mathematics - Complex Variables ,46A20 - Abstract
We describe the strong dual space $({\mathcal O}^s (D))^*$ for the space ${\mathcal O}^s (D) = H^s (D) \cap {\mathcal O} (D)$ of holomorphic functions from the Sobolev space $H^s(D)$, $s \in \mathbb Z$, over a bounded simply connected plane domain $D$ with infinitely differential boundary $\partial D$. We identify the dual space with the space of holomorhic functions on ${\mathbb C}^n\setminus \overline D$ that belong to $H^{1-s} (G\setminus \overline D)$ for any bounded domain $G$, containing the compact $\overline D$, and vanish at the infinity. As a corollary, we obtain a description of the strong dual space $({\mathcal O}_F (D))^*$ for the space ${\mathcal O}_F (D)$ of holomorphic functions of finite order of growth in $D$ (here, ${\mathcal O}_F (D)$ is endowed with the inductive limit topology with respect to the family of spaces ${\mathcal O}^s (D)$, $s \in \mathbb Z$). In this way we extend the classical Grothendieck-K{\"o}the-Sebasti\~{a}o e Silva duality for the space of holomorphic functions. more...
- Published
- 2024
Catalog
3. Lower semicontinuity of monotone functionals in the mixed topology on Cb.
- Author
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Nendel, Max
- Subjects
RIESZ spaces ,INTEGRAL representations ,CONTINUOUS functions ,FUNCTIONALS ,TOPOLOGY - Abstract
The main result of this paper characterises the continuity from below of monotone functionals on the space C b of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair (C b , ca) , where ca denotes the space of all countably additive signed Borel measures of finite variation. Hence lower semicontinuity in the mixed topology is for convex monotone maps C b → R equivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g. in the context of risk measures and superhedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities are studied. Moreover, a well-known characterisation of star-shaped risk measures on L ∞ is transferred to risk measures on C b . In a second step, the paper provides a characterisation of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on C b taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm-bounded sets. [ABSTRACT FROM AUTHOR] more...
- Published
- 2025
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4. Strong Duality and Solution Existence Under Minimal Assumptions in Conic Linear Programming.
- Author
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Luan, Nguyen Ngoc and Yen, Nguyen Dong
- Subjects
- *
CONVEX sets , *LINEAR programming , *TOPOLOGY , *VECTOR topology - Abstract
Conic linear programs in locally convex Hausdorff topological vector spaces are addressed in this paper. Solution existence for the dual problem, as well as solution existence for the primal problem, and strong duality, are proved under minimal regularity assumptions. Namely, to get the results and a Farkas-type theorem for infinite-dimensional conic linear inequalities, we employ the generalized Slater condition either for the primal problem or for the dual problem, as well as proper separation and the concept of quasi-regularity of convex sets. Illustrative examples are presented. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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5. The Orlicz–Pettis theorem for locally convex cones.
- Author
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Roth, Walter
- Abstract
We present a generalization of the classical Orlicz–Pettis theorem about subseries convergence in topological vector spaces. In preparation we review some aspects of the theory of locally convex cones, a generalization of locally convex topological vector spaces. We introduce conical extensions of the classical sequence spaces and a version of Schur’s theorem about weak and norm convergence of sequences in the extension of ℓ 1. Our version of the Orlicz–Pettis theorem refers to series of bounded elements of a locally convex cone. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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6. On linearisation and existence of preduals.
- Author
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Kruse, Karsten
- Abstract
We study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space X. Then we turn to the case that X = F (Ω) is a space of scalar-valued functions on a non-empty set Ω and characterise those among them which admit a special predual, namely a strong linearisation, i.e. there are a locally convex Hausdorff space Y, a map δ : Ω → Y and a topological isomorphism T : F (Ω) → Y b ′ such that T (f) ∘ δ = f for all f ∈ F (Ω) . [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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7. Refinements and Extensions of Some Strong Duality Theorems in Conic Linear Programming
- Author
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Luan, Nguyen Ngoc and Yen, Nguyen Dong
- Published
- 2024
- Full Text
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8. On the Predual of a Morrey–Lorentz Space and Its Applications to the Linear Calderón–Zygmund Operators.
- Author
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Dao, Nguyen Anh and Krantz, Steven G.
- Subjects
- *
CALDERON-Zygmund operator , *LINEAR operators , *COMMUTATORS (Operator theory) , *COMPACT operators , *COMMUTATION (Electricity) , *FACTORIZATION - Abstract
Our main purpose in this paper is to construct a predual of Morrey–Lorentz space via the block spaces, defined in [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1999, 28(1): 31–40]. As a direct application of duality, we obtain the Morrey–Lorentz boundedness of linear Calderón–Zygmund operators. Moreover, we study a weak Hardy factorization in terms of linear Calderón–Zygmund operators in Morrey–Lorentz spaces. As a consequence of this result, we obtain a characterization of functions in BMO(ℝn) through the boundedness of commutator [b, T], where T is a homogeneous Calderón–Zygmund operator. Finally, we prove a Morrey–Lorentz compactness characterization of [b, T]. Precisely, [b, T] is a compact operator on Morrey–Lorentz spaces if and only if b ∈ CMO(ℝn). [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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9. Generating functions in Riesz spaces.
- Author
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Azouzi, Youssef and Nasri, Youssef
- Subjects
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RIESZ spaces , *GENERATING functions , *FUNCTION spaces , *CONDITIONAL expectations - Abstract
We introduce and study the concept of generating function for natural elements in a Dedekind complete Riesz space equipped with a conditional expectation operator. This allows us to study discrete processes in a free-measure setting. In particular we improve a result obtained by Kuo, Vardy and Watson concerning Poisson approximation. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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10. A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces.
- Author
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Kalauch, Anke, Kuo, Wenchi, and Watson, Bruce A.
- Subjects
- *
RIESZ spaces , *CONDITIONAL expectations , *MARKOV processes , *MARTINGALES (Mathematics) - Abstract
We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [B.A. Watson, An Andô-Douglas type theorem in Riesz spaces with a conditional expectation, Positivity, 13 (2009), 543–558] and a Riesz-Frechet representation theorem for the T-strong dual, where T is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andô-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
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11. Concepts of almost periodicity and ergodic theorems in locally convex spaces.
- Author
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Amini, Fardin and Saeidi, Shahram
- Abstract
We wish to investigate mean ergodic theorems for generalizations of almost periodic functions on semigroups, as well as for semigroups of operators in the framework of locally convex spaces. Specially, we present functional characterizations of concepts of almost periodicity for vector-valued functions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
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12. Further inequalities involving the weighted geometric operator mean and the Heinz operator mean.
- Author
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Al-Subaihi, Ibrahim Ahmed and Raïssouli, Mustapha
- Subjects
- *
REAL numbers , *HILBERT space , *ENTROPY - Abstract
In this paper, we first investigate some inequalities involving the p-weighted geometric operator mean A ♯ p B = A 1 / 2 ( A − 1 / 2 B A − 1 / 2 ) p A 1 / 2 , where p ∈ [0, 1] is a real number and A, B are two positive invertible operators acting on a Hilbert space. As applications, we obtain some inequalities about the so-called Tsallis relative operator entropy. We also give some inequalities involving the Heinz operator mean. Our results refine some inequalities existing in the literature. In a second part, we construct iterative algorithms converging to A ♯ p B with a high rate of convergence. Some relationships involving A ♯ p B are deduced. Numerical examples illustrating the theoretical results are also discussed. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
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13. Bu’s theorem in the positive situation
- Author
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Bougoutaia Amar and Belacel Amar
- Subjects
banach lattice ,positive operators ,positive p−summing operators ,46a20 ,47a05 ,46b42 ,47b10 ,47b65 ,Mathematics ,QA1-939 - Abstract
In this paper, we valorize the relationship between positive p−summing operators and positive strongly q−summing operators using (Contemp. Math. 328, 145 − 149 (2003)).
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- 2022
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14. A duality result in locally convex cones.
- Author
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Dastouri, Amir and Ranjbari, Asghar
- Abstract
In this paper, we consider a locally convex cone (P , V) and verify the dual of (C o n v (P) , V ¯) the locally convex cone of the non-empty convex subsets of P . Under some semilattice conditions, we characterize the dual of C o n v (⋯ ⏟ n t i m e s (C o n v (P)) . [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
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15. Set-valued dynamic risk measures for processes and for vectors.
- Author
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Chen, Yanhong and Feinstein, Zachary
- Subjects
MEASUREMENT ,DEFINITIONS - Abstract
The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast to scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilise this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multi-portfolio time-consistency between set-valued risk measures for processes and vectors is provided. To accomplish this, an augmented definition for multi-portfolio time-consistency of set-valued risk measures for processes is proposed. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
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16. Domination and Kwapień’s factorization theorems for positive Cohen p–nuclear m–linear operators
- Author
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Bougoutaia Amar, Belacel Amar, and Hamdi Halima
- Subjects
banach lattice ,cohen positive strongly p–summing multilinear operators ,kwapień’s factorization theorem ,pietsch domination theorem ,positive p–summing operators ,tensor norm ,46a20 ,46a32 ,46b42 ,47a07 ,47b10 ,47b65 ,Mathematics ,QA1-939 - Abstract
In this paper, we introduce and study the concept of positive Cohen p-nuclear multilinear operators between Banach lattice spaces. We prove a natural analog to the Pietsch domination theorem for this class. Moreover, we give like the Kwapień’s factorization theorem. Finally, we investigate some relations with another known classes. more...
- Published
- 2021
- Full Text
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17. The logarithmic mean of two convex functionals
- Author
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Raïssouli Mustapha and Furuichi Shigeru
- Subjects
functional means ,logarithmic mean of convex functionals ,functional inequalities ,46n10 ,46a20 ,47a63 ,47n10 ,Mathematics ,QA1-939 - Abstract
The purpose of this paper is to introduce the logarithmic mean of two convex functionals that extends the logarithmic mean of two positive operators. Some inequalities involving this functional mean are discussed as well. The operator versions of the functional theoretical results obtained here are immediately deduced without referring to the theory of operator means. more...
- Published
- 2020
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18. Quasidensity: A Survey and Some Examples
- Author
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Simons, Stephen, Bauschke, Heinz H., editor, Burachik, Regina S., editor, and Luke, D. Russell, editor
- Published
- 2019
- Full Text
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19. Topological duals of locally convex function spaces.
- Author
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Pennanen, Teemu and Perkkiö, Ari-Pekka
- Abstract
This paper studies topological duals of locally convex function spaces that are natural generalizations of Fréchet and Banach function spaces. The dual is identified with the direct sum of another function space, a space of purely finitely additive measures and the annihilator of L ∞ . This allows for quick proofs of various classical as well as new duality results e.g. in Lebesgue, Musielak–Orlicz, Orlicz–Lorentz space and spaces associated with convex risk measures. Beyond Banach and Fréchet spaces, we obtain completeness and duality results in general paired spaces of random variables. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
- Full Text
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20. No-arbitrage concepts in topological vector lattices.
- Author
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Platen, Eckhard and Tappe, Stefan
- Subjects
RIESZ spaces ,STOCHASTIC analysis ,FUNCTION spaces ,BANACH spaces ,FINANCIAL markets - Abstract
We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA 1 and NA 1 may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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21. On a 'philosophical' question about Banach envelopes.
- Author
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Albiac, Fernando, Ansorena, José L., and Wojtaszczyk, Przemysław
- Abstract
We show how to construct non-locally convex quasi-Banach spaces X whose dual separates the points of a dense subspace of X but does not separate the points of X. Our examples connect with a question raised by Pietsch (Rev Mat Complut 22(1):209–226, 2009) and shed light into the unexplored class of quasi-Banach spaces with nontrivial dual which do not have sufficiently many functionals to separate the points of the space. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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22. Hedging of claims with physical delivery under convex transaction costs
- Author
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Pennanen, Teemu and Penner, Irina
- Subjects
Quantitative Finance - Pricing of Securities ,Mathematics - Functional Analysis ,Mathematics - Probability ,26B25 ,49J53 ,60G42 ,46A20 - Abstract
We study superhedging of contingent claims with physical delivery in a discrete-time market model with convex transaction costs. Our model extends Kabanov's currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer's robust no arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability. Combined with classical techniques of convex analysis, the closedness yields a dual characterization of premium processes that are sufficient to superhedge a given claim process. We also extend the fundamental theorem of asset pricing for general conical models. more...
- Published
- 2008
23. Superhedging in illiquid markets
- Author
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Pennanen, Teemu
- Subjects
Quantitative Finance - Pricing of Securities ,Mathematics - Optimization and Control ,Mathematics - Probability ,26B25 ,46A20 ,49J53 ,60G42 - Abstract
We study contingent claims in a discrete-time market model where trading costs are given by convex functions and portfolios are constrained by convex sets. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. We derive dual characterizations of superhedging conditions for contingent claim processes in a market without a cash account. The characterizations are given in terms of stochastic discount factors that correspond to martingale densities in a market with a cash account. The dual representations are valid under a topological condition and a weak consistency condition reminiscent of the ``law of one price'', both of which are implied by the no arbitrage condition in the case of classical perfectly liquid market models. We give alternative sufficient conditions that apply to market models with nonlinear cost functions and portfolio constraints. more...
- Published
- 2008
24. Recent Progress in Special Colombeau Algebras: Geometry, Topology, and Algebra
- Author
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Kunzinger, Michael
- Subjects
Mathematics - Functional Analysis ,Mathematics - Commutative Algebra ,46F30 ,46T30 ,46A20 ,16D25 ,53B20 - Abstract
Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of modules over the ring of generalized numbers, and algebraic aspects of Colombeau theory. Some open problems are given and directions of further research are outlined., Comment: 12 pages, contribution for conference "Generalized Functions 2007", Banach Center, Bedlewo, Poland more...
- Published
- 2007
25. On weakly extremal structures in Banach spaces
- Author
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Talponen, J.
- Subjects
Mathematics - Functional Analysis ,46B20 ,46A20 - Abstract
This paper deals with the interplay of the geometry of the norm and the weak topology in Banach spaces. Both dual and intrinsic connections between weak forms of rotundity and smoothness ared discussed. Weakly exposed points, weakly locally uniformly rotund spaces, smoothness, duality and the interplay of all the above are studied., Comment: 7 pages more...
- Published
- 2007
26. Time Consistent Dynamic Risk Processes, Cadlag Modification
- Author
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Bion-Nadal, Jocelyne
- Subjects
Mathematics - Probability ,Quantitative Finance - Risk Management ,46A20 ,91B30 ,91B70 - Abstract
Working in a continuous time setting, we extend to the general case of dynamic risk measures continuous from above the characterization of time consistency in terms of ``cocycle condition'' of the minimal penalty function. We prove also the supermartingale property for general time consistent dynamic risk measures. When the time consistent dynamic risk measure (continuous from above) is normalized and non degenerate, we prove, under a mild condition, that the dynamic risk process of any financial instrument has a cadlag modification. This condition is always satisfied in case of continuity from below., Comment: 23 pages more...
- Published
- 2006
27. Time consistency for scalar multivariate risk measures.
- Author
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Feinstein, Zachary and Rudloff, Birgit
- Subjects
TRANSACTION costs ,MARKETING costs ,SYSTEMIC risk (Finance) ,RISK perception - Abstract
In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a "moving scalarization" for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma and Zhang (2017), Kováčová and Rudloff (2021)). [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
- View/download PDF
28. On Stability and Weak-Star Stability of ε-Isometries.
- Author
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Chen, Xiaoling, Cheng, Lixin, and Zhang, Wen
- Abstract
Let X, Y be two Banach spaces, f : X → Y be an ε -isometry with f (0) = 0 for some ε ≥ 0 , and let Y f ≡ span ¯ f (X) . In this paper, we first introduce a notion of w ∗ -stability of an ε -isometry f. Then we show that stability of f implies its w ∗ -stability; the two notions of stability and w ∗ -stability coincide whenever X is a dual space and they are not equivalent in general. Making use of a recent sharp weak stability estimate of f, we then improve some known results. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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29. A duality proof of Tchakaloff's theorem
- Author
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Curto, Raul E. and Fialkow, Lawrence A.
- Subjects
Mathematics - Functional Analysis ,47A57 ,46A20 ,65D32 ,44A60 - Abstract
Tchakaloff's Theorem establishes the existence of a quadrature rule of prescribed degree relative to a positive, compactly supported measure that is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}^{d}$. Subsequent extensions were obtained by Mysovskikh and by Putinar. We provide new proofs and partial extensions of these results, based on duality techniques utilized by Stochel. We also obtain new uniqueness criteria in the Truncated Complex Moment Problem., Comment: 16 pages more...
- Published
- 2002
30. The strong Fatou property of risk measures
- Author
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Chen Shengzhong, Gao Niushan, and Xanthos Foivos
- Subjects
fatou property ,strong fatou property ,super fatou property ,dual representations ,law-invariant risk measures ,surplus-invariant risk measures ,inf-convolutions ,91g80 ,46e30 ,46a20 ,Science (General) ,Q1-390 ,Mathematics ,QA1-939 - Abstract
In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property. more...
- Published
- 2018
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31. On linear continuous operators between distinguished spaces Cp(X)
- Author
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Ka̧kol, Jerzy and Leiderman, Arkady
- Abstract
As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space C p (X) is distinguished if and only if X is a Δ -space. If there exists a linear continuous surjective mapping T : C p (X) → C p (Y) and C p (X) is distinguished, then C p (Y) also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator T : C p (X) → C p (Y) above is open? Secondly, we devote a special attention to concrete distinguished spaces C p ([ 1 , α ]) , where α is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping T : C p ([ 1 , α ]) → C p (Y) is given. We also observe that for every countable ordinal α all closed linear subspaces of C p ([ 1 , α ]) are distinguished, thereby answering an open question posed in Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of Δ -spaces we prove that a linear continuous surjection T : C p (X) → C k (X) w , where C k (X) w denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact X ⊂ R n ). [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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32. Set-valued risk measures as backward stochastic difference inclusions and equations.
- Author
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Ararat, Çağın and Feinstein, Zachary
- Subjects
STOCHASTIC difference equations ,STOCHASTIC differential equations ,DIFFERENCE equations ,DIFFERENTIAL inclusions - Abstract
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented via backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete-time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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33. Functional Covering Numbers.
- Author
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Artstein-Avidan, Shiri and Slomka, Boaz A.
- Abstract
We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality between separation and covering numbers. We provide analogues for various geometric inequalities on covering numbers, such as volume bounds, bounds connected with Hadwiger's conjecture, and inequalities about M-positions for geometric log-concave functions. In particular we get strong versions of M-positions for geometric log-concave functions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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34. Infinite-dimensional vector optimization and a separation theorem.
- Author
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Amahroq, Tijani and Oussarhan, Abdessamad
- Abstract
Using relative minimality solutions for vector optimization problems, we establish optimality conditions in terms of Karush–Kuhn–Tucker multipliers as well as Fritz-John multipliers. Moreover, we give a general separation theorem including both finite and infinite classical separation theorem. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
35. On closedness of convex sets in Banach lattices.
- Author
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Tantrawan, Made and Leung, Denny H.
- Subjects
BANACH lattices ,ORLICZ spaces ,TOPOLOGY - Abstract
Let X be a Banach lattice. A well-known problem arising from the theory of risk measures asks when order closedness of a convex set in X implies closedness with respect to the topology σ (X , X n ∼) , where X n ∼ is the order continuous dual of X. Motivated by the solution in the Orlicz space case, we introduce two relevant properties: the disjoint order continuity property (DOCP) and the order subsequence splitting property (OSSP). We show that when X is monotonically complete with OSSP and X n ∼ contains a strictly positive element, every order closed convex set in X is σ (X , X n ∼) -closed if and only if X has DOCP and either X or X n ∼ is order continuous. This in turn occurs if and only if either X or the norm dual X ∗ of X is order continuous. We also give a modular condition under which a Banach lattice has OSSP. In addition, we also give a characterization of X for which order closedness of a convex set in X is equivalent to closedness with respect to the topology σ (X , X uo ∼) , where X uo ∼ is the unbounded order continuous dual of X. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
36. On the matrix Monge–Kantorovich problem.
- Author
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CHEN, YONGXIN, GANGBO, WILFRID, GEORGIOU, TRYPHON T., and TANNENBAUM, ALLEN
- Subjects
- *
EUCLIDEAN distance , *MATRICES (Mathematics) , *QUANTUM mechanics - Abstract
The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
37. On closedness of law-invariant convex sets in rearrangement invariant spaces.
- Author
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Tantrawan, Made and Leung, Denny H.
- Abstract
This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space X . In particular, we show that order closedness, σ (X , X n ∼) -closedness, and σ (X , L ∞) -closedness of a law-invariant convex set in X are equivalent, where X n ∼ is the order continuous dual of X . We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
38. On the quasi-sure superhedging duality with frictions.
- Author
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Bayraktar, Erhan and Burzoni, Matteo
- Subjects
TRANSACTION costs ,FINANCIAL markets ,ARBITRAGE ,CONES - Abstract
We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modelled through solvency cones as in the original model of Kabanov (Finance Stoch. 3:237–248, 1999) adapted to the quasi-sure setup of Bouchard and Nutz (Ann. Appl. Probab. 25:823–859, 2015). Our approach allows removing the restrictive assumption of no arbitrage of the second kind considered in Bouchard et al. (Math. Finance 29:837–860, 2019) and showing the duality under the more natural condition of strict no arbitrage. In addition, we extend the results to models with portfolio constraints. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
39. Dual representations for systemic risk measures.
- Author
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Ararat, Çağın and Rudloff, Birgit
- Abstract
The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can represent bailout costs. More recently, a framework has been introduced, where institutions are supplied with capital allocations before aggregation. This yields an interpretation that is particularly useful for regulatory purposes. In each framework, the set of all feasible capital allocations leads to a multivariate risk measure. In this paper, we present dual representations for scalar systemic risk measures as well as for the corresponding multivariate risk measures concerning capital allocations. Our results cover both frameworks: aggregating after allocating and allocating after aggregation. As examples, we consider the aggregation mechanisms of the Eisenberg–Noe model as well as those of the resource allocation and network flow models. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
40. Topological rigidity as a monoidal equivalence.
- Author
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Poinsot, Laurent
- Subjects
- *
MATHEMATICAL equivalence , *TOPOLOGICAL algebras , *TENSOR products , *COMMUTATIVE rings , *ALGEBRA , *TOPOLOGY , *ALGEBRAIC topology , *ISOMORPHISM (Mathematics) - Abstract
A topological commutative ring is said to be rigid when for every set X, the topological dual of the X-fold topological product of the ring is isomorphic to the free module over X. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of "topologically free" modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and some coalgebras. In particular, we provide a description of its relationship with the standard duality between algebras and coalgebras, namely finite duality. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
41. Time-consistency of risk measures: how strong is such a property?
- Author
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Mastrogiacomo, Elisa and Rosazza Gianin, Emanuela
- Subjects
RISK ,AXIOMS ,CONVEXITY spaces ,MATHEMATICS - Abstract
Quite recently, a great interest has been devoted to time-consistency of risk measures in its different formulations (see Delbaen in Memoriam Paul-André Meyer, Lecture notes in mathematics, vol 1874, pp 215–258, 2006; Föllmer and Penner in Stat Decis 14(1):1–15, 2006; Bion-Nadal in Stoch Process Appl 119:633–654, 2009; Delbaen et al. in Finance Stoch 14(3):449–472, 2010; Laeven and Stadje in Math Oper Res 39:1109–1141, 2014, among many others). However, almost all the papers address to coherent or convex risk measures satisfying cash-additivity. In the present work, we study time-consistency for more general dynamic risk measures where either only cash-invariance or both cash-invariance and convexity are dropped. This analysis is motivated by the recent papers of El Karoui and Ravanelli (Math Finance 19:561–590, 2009) and Cerreia-Vioglio et al. (Math Finance 21(4):743–774, 2011) who discussed and weakened the axioms above by introducing cash-subadditivity and quasi-convexity. In particular, we investigate and discuss whether the notion of time-consistency is too restrictive, when considered in the general framework of quasi-convex and cash-subadditive risk measures. Finally, we provide some conditions guaranteeing time-consistency in this more general framework. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
42. A local Hahn–Banach theorem and its applications.
- Author
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Gao, Niushan, Leung, Denny H., and Xanthos, Foivos
- Abstract
An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a "local" version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014). [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
43. Cohen positive strongly p-summing and p-convex multilinear operators.
- Author
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Bougoutaia, Amar and Belacel, Amar
- Subjects
BANACH lattices ,POSITIVE operators ,MATHEMATICS - Abstract
The aim of this work is to give and study the notion of Cohen positive p-summing multilinear operators. We prove a natural analog of the Pietsch domination theorem for these classes and characterize their conjugates. As an application, we generalize a result due to Bu and Shi (J. Math. Anal. Appl. 401:174–181, 2013), and we compare this class with the class of multiple p-convex m-linear operators. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
44. Weak compactness in locally convex cones.
- Author
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Motallebi, M. R.
- Subjects
CONES ,TOPOLOGY - Abstract
Using the coarsest weak topologies, we present the necessary and sufficient conditions for the weak upper, lower and symmetric compactness of subsets in cones. This leads us to investigate the weakly compact subsets in product cones and discuss the X-topologies of the weakly compact subsets on direct sum cones. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
45. Two Classes of Metrizable Spaces -Invariant
- Author
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López-Pellicer, Manuel, Ferrando, Juan Carlos, editor, and López-Pellicer, Manuel, editor
- Published
- 2014
- Full Text
- View/download PDF
46. Stability Characterizations of ∈-isometries on Certain Banach Spaces.
- Author
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Cheng, Li Xin and Sun, Long Fa
- Subjects
- *
BANACH spaces , *SUBSPACES (Mathematics) , *STABILITY theory , *ISOMETRICS (Mathematics) , *INDECOMPOSABLE modules - Abstract
Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-isometry f: X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ∈-isometry to be stable in assuming that N is w*-closed in Y*. Making use of this result, we improve several known results including Figiel's theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span¯[f(x)] contains a complemented linear isometric copy of X; Moreover, if X = Y, then for every ∈-isometry f : X → X, there exists a surjective linear isometry S : X → X such that f − S is uniformly bounded by 2∈ on X. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
- View/download PDF
47. The Geometry of m-Hyperconvex Domains.
- Author
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Åhag, Per, Czyż, Rafał, and Hed, Lisa
- Abstract
We study the geometry of m-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every m-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
- Full Text
- View/download PDF
48. Fatou closedness under model uncertainty.
- Author
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Maggis, Marco, Meyer-Brandis, Thilo, and Svindland, Gregor
- Subjects
FATOU theorems ,MATHEMATICAL models of uncertainty ,MONOTONE operators ,CONVEX sets ,BANACH lattices - Abstract
We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of P-quasisure bounded random variables, where P is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
- Full Text
- View/download PDF
49. Duality for unbounded order convergence and applications.
- Author
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Gao, Niushan, Leung, Denny H., and Xanthos, Foivos
- Subjects
DUALITY (Logic) ,MATHEMATICAL bounds ,STOCHASTIC convergence ,GENERALIZATION ,BANACH lattices - Abstract
Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) Xuo∼
of a Banach lattice X and identify it as the order continuous part of the order continuous dual Xn∼ . The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the Fenchel-Moreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance. [ABSTRACT FROM AUTHOR] more... - Published
- 2018
- Full Text
- View/download PDF
50. Isometric representations of weighted spaces of little Lipschitz functions.
- Author
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Jiménez-Vargas, A. and Rueda, P.
- Abstract
Given a compact pointed metric space
X and a weightv on the complement of the diagonal set in X×X, we prove that the Banach space lipv(X) of all weighted little Lipschitz scalar-valued functions on X vanishing at the basepoint, equipped with the weighted Lipschitz norm, embeds almost isometrically into c0. This result has many consequences on the structure of those Banach spaces and their duals. Moreover, we prove that this isomorphism can never be an isometric embedding whenever X is a T-balanced subset containing 0 and compact for some metrizable topology of a complex Banach space and v is a radial 0-weight. [ABSTRACT FROM AUTHOR] more...- Published
- 2018
- Full Text
- View/download PDF
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