Representations of set-valued risk measures defined on the l-tensor product of Banach lattices

We obtain a representation for set-valued risk measures which are defined on the completed \(l\) -tensor product \(E\widetilde{\otimes }_l G\) of Banach lattices \(E\) and \(G\) . This representation extends known representations for set-valued risk measures defined on Bochner spaces \(L^p(\mathbb {P}, \mathbb {R}^d)\) of \(p\) -integrable functions with values in \(\mathbb {R}^d\) .     

Fatou closedness under model uncertainty

We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of \({\mathcal P}\)-quasisure bounded random variables, where \({\mathcal P}\) 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.     

On the representation of tight functionals as integrals

We consider a vector lattice $\mathcal L $ of bounded real continuous functions on a topological space $X$ that separates the points of $X$ and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on $\mathcal L $ can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.     

Addendum to: Entropic Value-at-Risk: A New Coherent Risk Measure

This short addendum consists of two sections. The first provides proofs that were omitted in Ahmadi-Javid (J. Optim. Theory Appl., 2012) for the sake of brevity, and also demonstrates that the dual representation of the entropic value-at-risk, which is given in Ahmadi-Javid (J. Optim. Theory Appl., 2012) for the case of bounded random variables, holds for all random variables whose moment-generating functions exist everywhere. The second section provides a few corrections.     

Second-order properties of risk concentrations without the condition of asymptotic smoothness

For the purpose of risk management, the quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. Consider a portfolio of n independent and identically distributed loss random variables with a common survival function $\overline {F}$ possessing the property of second-order regular variation. Under the additional assumption that $\overline {F}$ is asymptotically smooth, Degen et al. (Insur Math Econ 46:541–546, 2010) and Mao et al. (Insur Math Econ 51:449–456, 2012) derived second-order approximations of the risk concentrations based on the risk measures of Value-at-Risk and conditional tail expectation, respectively. In this paper, we remove the assumption of the asymptotic smoothness, and reestablish the second-order approximations of these two risk concentrations.     

Extensions of the Borel–Cantelli lemma in general measure spaces

In this paper, an important bilateral inequality for a sequence of nonnegative measurable functions on a measure space \((S,\mathcal {B}_S,\mu )\) is obtained, and some sufficient conditions for \(\mu \left( \limsup \limits _{n\rightarrow \infty }A_n\right) =\mu (S)\) are given. In addition, a weighted version of the Borel–Cantelli Lemma on the measure space is obtained. Our results generalize the corresponding ones for bounded random sequences to the case of unbounded measurable functions.     

On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\) , we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\) -matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szegö kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.     

A sequence space representation of L. Schwartz’ space {{\mathcal{O}_{C}}}

We derive a representation of the isomorphic spaces ${\mathcal{O}_{C}}$ of very slowly increasing functions and ${\mathcal{O}_{M}'}$ of very rapidly decreasing distributions as a completed topological tensor product of sequence spaces. In order to describe this completed topological tensor product as a space of double sequences, we construct a representation as an inductive limit of vector valued sequence spaces. Moreover we compare the representations of ${\mathcal{O}_{C}}$ and ${\mathcal{O}_{M}}$ .     

Independence Under the G-Expectation Framework

We show that, for two non-trivial random variables \(X\) and \(Y\) under a sublinear expectation space, if \(X\) is independent from \(Y\) and \(Y\) is independent from \(X\) , then \(X\) and \(Y\) must be maximally distributed.     

A structural characterization of numéraires of convex sets of nonnegative random variables

We introduce the concept of numéraire s of convex sets in ${L^0_{+}}$ , the nonnegative orthant of the topological vector space L ⁰ of all random variables built over a probability space. A necessary and sufficient condition for an element of a convex set ${\mathcal{C} \subseteq L^0_{+}}$ to be a numéraire of ${\mathcal{C}}$ is given, inspired from ideas in financial mathematics.     

Classical Limit of the Scaled Elliptic Algebra \mathcal{A} ħ,η (\widetilde{\mathfrak{s}\mathfrak{l}}_2)

The classical limit of the scaled elliptic algebra $\mathcal{A}$ ?,η ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $\mathcal{A}$ q,p ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is also briefly presented.     

A Family of Measures of Noncompactness in the Space {L^1_{\text{loc}}(\mathbb{R}_+)} and its Application to Some Nonlinear Volterra Integral Equation

The aim of this paper is to study a new family of measures of noncompactness in the space ${L^1_{\text{loc}}(\mathbb{R}_+)}$ consisting of all real functions locally integrable on ${\mathbb{R}_+}$ , equipped with a suitable topology. As an example of applications of the technique associated with that family of measures of noncompactness, we study the existence of solutions of a nonlinear Volterra integral equation in the space ${L^1_{\text{loc}}(\mathbb{R}_+)}$ . The obtained result generalizes several ones obtained earlier with help of other methods.     

Norm-Constrained Determinantal Representations of Multivariable Polynomials

For every multivariable polynomial $p$ , with $p(0)=1$ , we construct a determinantal representation, $ p=\det (I - K Z )$ , where $Z$ is a diagonal matrix with coordinate variables on the diagonal and $K$ is a complex square matrix. Such a representation is equivalent to the existence of $K$ whose principal minors satisfy certain linear relations. When norm constraints on $K$ are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial $q$ , $q(0)=0,$ satisfies the von Neumann inequality, then $1-q$ admits a determinantal representation with $K$ a contraction. On the other hand, every determinantal representation with a contractive $K$ gives rise to a rational inner function in the Schur–Agler class.     

Semicontinuous limits of nets of continuous functions

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space $X$ that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions $\mathcal{C }(X)$ is the set of upper semicontinuous functions on $X$ , and (2) the pointwise convergence of a net in $\mathcal{C }(X)$ to an upper semicontinuous limit automatically ensures convergence in this finer topology.     

Riesz completions,functional representations,and anti-lattices

We show that the Riesz completion of an Archimedean partially ordered vector space $X$ with unit can be represented as a norm dense Riesz subspace of the smallest functional representation of $X.$ This yields a convenient way to find the Riesz completion. To illustrate the method, the Riesz completions of spaces ordered by Lorentz cones, cones of symmetric positive semi-definite matrices, and polyhedral cones are determined. We use the representation to analyse the existence of non-trivial disjoint elements and link the absence of such elements to the notion of anti-lattice. One of the results is a geometric condition on the dual cone of a finite dimensional partially ordered vector space $X$ that ensures that $X$ is an anti-lattice.     

Spinorial representation of surfaces into 4-dimensional space forms

In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of minimal surfaces. We also obtain as particular cases the spinorial characterizations of surfaces in $\mathbb R ^3$ and in $S^3$ given by Friedrich and by Morel.     

A variance-based method to rank input variables of the Mesh Adaptive Direct Search algorithm

The Mesh Adaptive Direct Search algorithm (Mads) algorithm is designed for nonsmooth blackbox optimization problems in which the evaluation of the functions defining the problems are expensive to compute. The Mads algorithm is not designed for problems with a large number of variables. The present paper uses a statistical tool based on variance decomposition to rank the relative importance of the variables. This statistical method is then coupled with the Mads algorithm so that the optimization is performed either in the entire space of variables or in subspaces associated with statistically important variables. The resulting algorithm is called Stats-Mads and is tested on bound constrained test problems having up to 500 variables. The numerical results show a significant improvement in the objective function value after a fixed budget of function evaluations.     

Frequent Hypercyclicity of Random Entire Functions for the Differentiation Operator

We study the random entire functions defined as power series \(f(z) = \sum _{n=0}^\infty (X_n/n!) z^n\) with independent and identically distributed coefficients \((X_n)\) and show that, under very weak assumptions, they are frequently hypercyclic for the differentiation operator \(D: H({\mathbb {C}}) \rightarrow H({\mathbb {C}}),\,f \mapsto Df = f'\) . This gives a very simple probabilistic construction of \(D\) -frequently hypercyclic functions in \(H({\mathbb {C}})\) . Moreover we show that, under more restrictive assumptions on the distribution of the \((X_n)\) , these random entire functions have a growth rate that differs from the slowest growth rate possible for \(D\) -frequently hypercyclic entire functions at most by a factor of a power of a logarithm.