Time consistency for scalar multivariate risk measures

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)).

Scalar Multivariate Risk Measures with a Single Eligible Asset

In this paper, we present results on scalar risk measures in markets with transaction costs. Such risk measures are defined as the minimal capital requirements in the cash asset. First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variables as (equivalent) martingale measures and prices consistent with the market model. Then, these dual representations are used to obtain the main results of this paper on time consistency for scalar risk measures in markets with frictions. It is well known from the superhedging risk measure in markets with transaction costs that the usual scalar concept of time consistency is too strong and not satisfied. We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to the underlying risk measure with a fixed consistent pricing process. Several examples are given, with special emphasis on the superhedging risk measure.

Mixed scalar-current bootstrap in three dimensions

We study the mixed system of correlation functions involving a scalar field charged under a global U(1) symmetry and the associated conserved spin-1 current Jμ. Using numerical bootstrap techniques we obtain bounds on new observables not accessible in the usual scalar bootstrap. We then specialize to the O(2) model and extract rigorous bounds on the three-point function coefficient of two currents and the unique relevant scalar singlet, as well as those of two currents and the stress tensor. Using these results, and comparing with a quantum Monte Carlo simulation of the O(2) model conductivity, we give estimates of the thermal one-point function of the relevant singlet and the stress tensor. We also obtain new bounds on operators in various sectors.

Balancing sets of vectors

Let n be an arbitrary integer, let p be a prime factor of n . Denote by ω1 the pth primitive unity root, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\omega _1 : = e^{\tfrac{{2\pi i}} {p}}$$ \end{document}.Define ωi ≔ ω1i for 0 ≦ i ≦ p − 1 and B ≔ {1, ω1 , …, ωp −1 } n ⊆ ℂ n .Denote by K ( n; p ) the minimum k for which there exist vectors ν1 , …, νk ∈ B such that for any vector w ∈ B , there is an i , 1 ≦ i ≦ k , such that νi · w = 0, where ν · w is the usual scalar product of ν and w .Gröbner basis methods and linear algebra proof gives the lower bound K ( n; p ) ≧ n ( p − 1).Galvin posed the following problem: Let m = m ( n ) denote the minimal integer such that there exists subsets A1 , …, Am of {1, …, 4 n } with | Ai | = 2 n for each 1 ≦ i ≦ n , such that for any subset B ⊆ [4 n ] with 2 n elements there is at least one i , 1 ≦ i ≦ m , with Ai ∩ B having n elements. We obtain here the result m ( p ) ≧ p in the case of p > 3 primes.

PHANTOM THICK BRANE IN 5D BULK

A model of a thick brane in 5D bulk supported by two phantom scalar fields is considered. The comparison with a thick brane supported by two usual scalar fields is carried out. The distinctions between a thick brane supported by one usual scalar field and our model have been pointed out.

NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED

Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

C*-Convexity and Matricial Ranges

C* -convex sets in matrix algebras are convex sets of matrices in which matrix-valued convex coefficients are admitted along with the usual scalar-valued convex coefficients. A Carathéodory-type theorem is developed for C*-convex hulls of compact sets of matrices, and applications of this theorem are given to the theory of matricial ranges. If T is an element in a unital C*-algebra , then for every n ∈ N, the n x n matricial range Wn(T) of T is a compact C* -convex set of n x n matrices. The basic relation W1(T) = conv σ-(T) is well known to hold if T exhibits the normal-like quality of having the spectral radius of β T + μ 1 coincide with the norm ||β T + μ 1|| for every pair of complex numbers β and μ. An extension of this relation to the matrix spaces is given by Theorem 2.6: Wn (T) is the C*-convex hull of the n x n matricial spectrum σn(T) of T if, for every B,M ∈ ℳn, the norm of T ⊗ B + 1 ⊗ M in ⊗ ℳn is the maximum value in {||∧⊗B + 1 ⊗M|| : Λ ∈ σn (T)}. The spatial matricial range of a Hilbert space operator is the analogue of the classical numerical range, although it can fail to be convex if n > 1. It is shown in § 3 that if T has a normal dilation N with σ (N) ⊂ σ (T), then the closure of the spatial matricial range of T is convex if and only if it is C*-convex.

Clifford algebras and vector-valued rational forms. I

We demonstrate how Clifford algebras offer a framework for the construction of vector-valued rational forms possessing features of the usual scalar theory, including three-term recurrence relations for continued fractions. The price for this advantage is that the Moore–Penrose generalized inverse is replaced by the multiplicative group inverse of a Clifford algebra. However, the connection between the new vector-valued rational forms and generalized inverse rational forms is a close one; in fact, the two forms are identical for real analytic data.