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Foivos Xanthos

Publications and source records attributed to Foivos Xanthos.

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

A universal approximation theorem and its applications to vector lattice theory

A classical result in approximation theory states that for any continuous function \( φ: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{φ\circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if \( φ\) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improve the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}.

math.FA↗
arXiv2025

Pervasiveness of $\mathcal{L}^r(E,F)$ in $\mathcal{L}^r(E,F^δ)$

Let $E, F$ be Archimedean Riesz spaces, and let $F^δ$ denote an order completion of $F$. In this note, we provide necessary conditions under which the space of regular operators $\mathcal{L}^r(E, F)$ is pervasive in $\mathcal{L}^r(E, F^δ)$. Pervasiveness of $\mathcal{L}^r(E, F)$ in $\mathcal{L}^r(E, F^δ)$ implies that the Riesz completion of $ \mathcal{L}^r(E, F)$ can be realized as a Riesz subspace of $ \mathcal{L}^r(E, F^δ$. It also ensures that the regular part of the space of order continuous operators $\mathcal{L}^{oc}(E, F)$ forms a band of $\mathcal{L}^r(E, F)$. Furthermore, the positive part $T^+$ of any operator $T \in \mathcal{L}^r(E, F)$, provided it exists, is given by the Riesz-Kantorovich formula. The results apply in particular to cases where $E = \ell_0^{\infty}$, $E = c$, or $F$ is atomic, and they provide solutions to some problems posed in [3] and [16].

math.FA↗
arXiv2024

A note on continuity and asymptotic consistency of measures of risk and variability

In this short note, we show that every convex, order bounded above functional on a Frechet lattice is automatically norm continuous. This improves a result in \cite{RS06} and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in \cite{KSZ14}.

q-fin.RM↗
arXiv2020

Stability properties of Haezendonck-Goovaerts premium principles

We investigate a variety of stability properties of Haezendonck-Goovaerts premium principles on their natural domain, namely Orlicz spaces. We show that such principles always satisfy the Fatou property. This allows to establish a tractable dual representation without imposing any condition on the reference Orlicz function. In addition, we show that Haezendonck-Goovaerts principles satisfy the stronger Lebesgue property if and only if the reference Orlicz function fulfills the so-called $Δ_2$ condition. We also discuss (semi)continuity properties with respect to $Φ$-weak convergence of probability measures. In particular, we show that Haezendonck-Goovaerts principles, restricted to the corresponding Young class, are always lower semicontinuous with respect to the $Φ$-weak convergence.

q-fin.MF↗
arXiv2020

On the extension property of dilatation monotone risk measures

Let $\mathcal{X}$ be a subset of $L^1$ that contains the space of simple random variables $\mathcal{L}$ and $ρ: \mathcal{X} \rightarrow (-\infty,\infty]$ a dilatation monotone functional with the Fatou property. In this note, we show that $ρ$ extends uniquely to a $σ(L^1,\mathcal{L})$ lower semicontinuous and dilatation monotone functional $\overlineρ: L^1 \rightarrow (-\infty,\infty]$. Moreover, $\overlineρ$ preserves monotonicity, (quasi)convexity, and cash-additivity of $ρ$. Our findings complement recent extension results for quasiconvex law-invariant functionals proved in [17,20]. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to $L^1$ that retains the robust representations obtained in [4,6].

q-fin.MF↗
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

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↗
arXiv2018

The strong Fatou property of risk measures

In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], 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 $\mathcal{X}$ with the strong Fatou property is $σ(\mathcal{X},L^\infty)$ 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.

q-fin.RM↗
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↗
arXiv2016

Option spanning beyond $L_p$-models

\begin{abstract} The aim of this paper is to study the spanning power of options in a static financial market that allows non-integrable assets. Our findings extend and unify the results in [8,9,18] for $L_p$-models. We also apply the spanning power properties to the pricing problem. In particular, we show that prices on call and put options of a limited liability asset can be uniquely extended by arbitrage to all marketed contingent claims written on the asset.

q-fin.MF↗
arXiv2016

On the C-property and $w^*$-representations of risk measures

We identify a large class of Orlicz spaces $X$ for which the topology $σ(X,X_n^\sim)$ fails the C-property introduced in [7]. We also establish a variant of the C-property and use it to prove a $w^*$-representation theorem for proper convex increasing functionals on dual Banach lattices that satisfy a suitable version of Delbaen's Fatou property. Our results apply, in particular, to risk measures on all Orlicz spaces over $[0,1]$ which is not $L_1[0,1]$.

q-fin.MF↗
arXiv2015

Uo-convergence and its applications to Cesàro means in Banach lattices

A net $(x_α)$ in a vector lattice $X$ is said to uo-converge to $x$ if $|x_α-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].

math.FA↗
arXiv2015

Spaces of regular abstract martingales

In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm.

math.FA↗
arXiv2013

A version of Kalton's theorem for the space of regular operators

In this note we extend some recent results in the space of regular operators. In particular, we provide the following Banach lattice version of a classical result of Kalton: Let $E$ be an atomic Banach lattice with an order continuous norm and $F$ a Banach lattice. Then the following are equivalent: (i) $L^r(E,F)$ contains no copy of $\ell_\infty$, \,\, (ii) $L^r(E,F)$ contains no copy of $c_0$, \,\, (iii) $K^r(E,F)$ contains no copy of $c_0$, \,\, (iv) $K^r(E,F)$ is a (projection) band in $L^r(E,F)$, \,\, (v) $K^r(E,F)=L^r(E,F)$.

math.FA↗