Searcharxiv⌕ Search

arXiv subjects

Denny H. Leung

Publications and source records attributed to Denny H. Leung.

At least 19 recordsLinked to original sources
arXiv2026

On Prudence of Risk Measures

Prudence is a stability property of risk functionals recently introduced by Wang and Zitikis and subsequently studied by Amarante and Liebrich. In this paper, we first establish general relationships between prudence and other stability properties, showing, in particular, that weak prudence and prudence coincide for a broad class of convex, law-invariant functionals. We then prove that prudence is preserved by cash-additive hulls of star-shaped functionals under a simple asymptotic condition, and by inf-convolutions of convex, cash-additive, law-invariant prudent functionals. Our results provide general methods for constructing prudent risk measures from existing prudent functionals.

q-fin.RM↗
arXiv2026

Banach lattices with upper $p$-estimates: Renorming and factorization

The notions of $p$-convexity and concavity are fundamental tools for studying Banach lattices, as they partition the class of Banach lattices into a scale of spaces with $L_p$-like properties. Upper and lower $p$-estimates provide a refinement of this scale, modeled by the Lorentz spaces $L_{p,\infty}$ and $L_{p,1}$, respectively. In this article, we provide a comprehensive treatment of Banach lattices with upper $p$-estimates. In particular, we show that many well-known theorems about $p$-convex Banach lattices have analogues in the upper $p$-estimate setting, including the ability to represent all such spaces inside of infinity sums of model spaces, to canonically factor the convex operators and identify their associated operator ideals, as well as to give a precise description of the free objects and push-outs. Proving these results is far from straightforward and will require the development of a variety of new tools that avoid convexification and concavification procedures. In fact, we will identify many fundamental differences between the theories of $p$-convexity and upper $p$-estimates, particularly with regards to isometric problems and renormings.

math.FA↗
arXiv2022

The order-type Banach-Saks properties

The study of the Banach-Saks property in Banach spaces has a long and illustrious history. Of late, motivated by applications in financial mathematics, interest has arisen in the Banach-Saks type properties with respect to order convergence. This paper presents a study of order Banach-Saks properties in Banach function spaces, and in particular in rearrangement invariant spaces. Among the results obtained, we provide some sufficient conditions for the (weak) order Banach-Saks property. We also characterize the (weak) order Banach-Saks property in Orlicz spaces. It is also shown that the (weak) order Banach-Saks property is equivalent to its hereditary version.

math.FA↗
arXiv2021

Do law-invariant linear functionals collapse to the mean?

In this note, we show that, on a wide range of rearrangement-invariant spaces, a law-invariant bounded linear functional is a scalar multiple of the expectation. We also construct a rearrangement-invariant space on which this property fails.

math.FA↗
arXiv2020

Nonlinear biseparating maps

An additive map $T$ acting between spaces of vector-valued functions is said to be biseparating if $T$ is a bijection so that $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. Note that an additive bijection retains $\mathbb{Q}$-linearity. For a general nonlinear map $T$, the definition of biseparating given above turns out to be too weak to determine the structure of $T$. In this paper, we propose a revised definition of biseparating maps for general nonlinear operators acting between spaces of vector-valued functions, which coincides with the previous definition for additive maps. Under some mild assumptions on the function spaces involved, it turns out that a map is biseparating if and only if it is locally determined. We then delve deeply into some specific function spaces -- spaces of continuous functions, uniformly continuous functions and Lipschitz functions -- and characterize the biseparating maps acting on them. As a by-product, certain forms of automatic continuity are obtained. We also prove some finer properties of biseparating maps in the cases of uniformly continuous and Lipschitz functions.

math.FA↗
arXiv2019

On closedness of law-invariant convex sets in rearrangement invariant spaces

This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $\mathcal{X}$. In particular, we show that order closedness, $σ(\mathcal{X},\mathcal{X}_n^\sim)$-closedness and $σ(\mathcal{X},L^\infty)$-closedness of a law-invariant convex set in $\mathcal{X}$ are equivalent, where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property.

q-fin.RM↗
arXiv2019

On local convexity in $\mathbb{L}^0$ and switching probability measures

In the paper, we investigate the following fundamental question. For a set $\mathcal{K}$ in $\mathbb{L}^0(\mathbb{P})$, when does there exist an equivalent probability measure $\mathbb{Q}$ such that $\mathcal{K}$ is uniformly integrable in $\mathbb{L}^1(\mathbb{Q})$. Specifically, let $\mathcal{K}$ be a convex bounded positive set in $\mathbb{L}^1(\mathbb{P})$. Kardaras [6] asked the following two questions: (1) If the relative $\mathbb{L}^0(\mathbb{P})$-topology is locally convex on $\mathcal{K}$, does there exist $\mathbb{Q}\sim \mathbb{P}$ such that the $\mathbb{L}^0(\mathbb{Q})$- and $\mathbb{L}^1(\mathbb{Q})$-topologies agree on ${\mathcal{K}}$? (2) If $\mathcal{K}$ is closed in the $\mathbb{L}^0(\mathbb{P})$-topology and there exists $\mathbb{Q}\sim \mathbb{P}$ such that the $\mathbb{L}^0(\mathbb{Q})$- and $\mathbb{L}^1(\mathbb{Q})$-topologies agree on $\mathcal{K}$, does there exist $\mathbb{Q}'\sim \mathbb{P}$ such that $\mathcal{K}$ is $\mathbb{Q}'$-uniformly integrable? In the paper, we show that, no matter $\mathcal{K}$ is positive or not, the first question has a negative answer in general and the second one has a positive answer. In addition to answering these questions, we establish probabilistic and topological characterizations of existence of $\mathbb{Q}\sim\mathbb{P}$ satisfying these desired properties. We also investigate the peculiar effects of $\mathcal{K}$ being positive.

math.PR↗
arXiv2018

On closedness of convex sets in Banach lattices

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 $σ(X,X_n^\sim)$, where $X_n^\sim$ 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 $X_n^\sim$ contains a strictly positive element, every order closed convex set in $X$ is $σ(X,X_n^\sim)$-closed if and only if $X$ has $DOCP$ and either $X$ or $X_n^\sim$ 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 $σ(X,X_{uo}^\sim)$, where $X_{uo}^\sim$ is the unbounded order continuous dual of $X$.

math.FA↗
arXiv2018

A Local Hahn-Banach Theorem and Its Applications

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 [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8].

math.FA↗
arXiv2017

Fatou Property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can be always achieved under the assumption of law-invariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka's representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipović and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.

q-fin.RM↗
arXiv2017

Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures

Let $(Φ,Ψ)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^Φ$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^Φ$ characterizes closedness with respect to the topology $σ(L^Φ,L^Ψ)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^Φ$, order closedness and $σ(L^Φ,L^Ψ)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $σ(L^Φ,L^Ψ)$-closedness of convex sets in $L^Φ$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $σ(L^Φ,L^Ψ)$; that is, a convex set is $σ(L^Φ,L^Ψ)$-closed if and only if it is closed with respect to the bounded-$σ(L^Φ,L^Ψ)$ topology. As a result, we show that order closedness and $σ(L^Φ,L^Ψ)$-closedness of convex sets in $L^Φ$ are equivalent if and only if either $Φ$ or $Ψ$ satisfies the $Δ_2$-condition. Using this, we prove the surprising result that: \emph{If (and only if) $Φ$ and $Ψ$ both fail the $Δ_2$-condition, then there exists a coherent risk measure on $L^Φ$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^Φ, L^Ψ)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^Φ,H^Ψ)$.

q-fin.MF↗
arXiv2017

Duality for unbounded order convergence and applications

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) $X_{uo}^\sim$ of a Banach lattice $X$ and identify it as the order continuous part of the order continuous dual $X_n^\sim$. 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.

math.FA↗
arXiv2017

Smallest order closed sublattices and option spanning

Let $Y$ be a sublattice of a vector lattice $X$. We consider the problem of identifying the smallest order closed sublattice of $X$ containing $Y$. It is known that the analogy with topological closure fails. Let $\overline{Y}^o$ be the order closure of $Y$ consisting of all order limits of nets of elements from $Y$. Then $\overline{Y}^o$ need not be order closed. We show that in many cases the smallest order closed sublattice containing $Y$ is in fact the second order closure $\overline{\overline{Y}^o}^o$. Moreover, if $X$ is a $σ$-order complete Banach lattice, then the condition that $\overline{Y}^o$ is order closed for every sublattice $Y$ characterizes order continuity of the norm of $X$. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset.

math.FA↗
arXiv2017

On ordinal ranks of Baire class functions

The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidnyánszky to Baire class $ξ$ functions for any countable ordinal $ξ\geq1$. In this paper, we answer two of the questions raised by them in their paper (Ranks on the Baire class $ξ$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143). Specifically, we show that for any countable ordinal $ξ\geq1,$ the ranks $β_ξ^{\ast}$ and $γ_ξ^{\ast}$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $β$ is not essentially multiplicative, we investigate further the behavior of this rank with respect to products. We characterize the functions $f$ so that $β(fg)\leq ω^ξ$ whenever $β(g)\leqω^ξ$ for any countable ordinal $ξ.$

math.FA↗
arXiv2016

All 2-positive linear maps from M3 to M3 are decomposable

Following an idea of Choi, we obtain a decomposition theorem for k-positive linear maps from Mm to Mn, where 2<=k<min{m,n}. As a consequence, we give an affirmative answer to Kye's conjecture (also solved independently by Choi) that every 2-positive linear map from M3 to M3 is decomposable.

math-ph↗
arXiv2015

Persistence of Banach lattices under nonlinear order isomorphisms

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection between them that preserves order. We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to c_0, and for Banach lattices that contain a closed sublattice order isomorphic to c_0.

math.FA↗
arXiv2014

Nonlinear order isomorphisms on function spaces

Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets of $C(X)$ and $C(Y)$ respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between $A(X)$ and $A(Y)$. Under some general assumptions on $A(X)$ and $A(Y)$, and when $X$ and $Y$ are compact Hausdorff, it is shown that existence of an order isomorphism between $A(X)$ and $A(Y)$ gives rise to an associated homeomorphism between $X$ and $Y$. This generalizes a classical result of Kaplansky concerning linear order isomorphisms between $C(X)$ and $C(Y)$ for compact Hausdorff $X$ and $Y$. The class of near vector lattices is introduced in order to extend the result further to noncompact spaces $X$ and $Y$. The main applications lie in the case when $X$ and $Y$ are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, "local" and "bounded local" versions of these spaces, characterizations of when spaces of one type can be order isomorphic to spaces of another type are obtained.

math.FA↗