Dynamic risk measures under model uncertainty

In this paper we consider the notion of dynamic risk measures, which we will motivate as a reasonable tool in risk management. It is possible to reformulate an example of such a risk measure in terms of the value functions of a Markov decision model (MDM). Based on this observation the model is generalized to a setting with incomplete information about the risk distribution which can be seen as model uncertainty. This issue can be incorporated in the dynamic risk measure by extending the MDM to a Bayesian decision model. Moreover, it is possible to discuss the effect of model uncertainty on the risk measure in binomial models. All investigations are illustrated by a simple but useful coin tossing game proposed by Artzner and by the classic Cox–Ross–Rubinstein model.     

Parity in partition identities

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f ₀(q) along with a new proof of the Hecke-type series representation.     

A note on blocks of skeleton tolerances

We provide a full characterization of lattices which can be blocks of the skeleton tolerance relation of a finite lattice. Moreover, we formulate a necessary condition for a lattice to be such a block in the case of finite distributive lattices with at most k-dimensional maximal boolean intervals.     

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

     

Triple Homomorphisms of Perfect Lie Algebras

Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.     

Eta products of weight one,and decomposition of primes in non-abelian extensions

This article is a slightly extended and revised version of a conference talk at “Arithmetik an der A7” in Würzburg, June 23rd, 2017. We present a conjecture on the coincidence of Hecke theta series of weight 1 on three distinct quadratic fields. Then we discuss a special instance of the Deligne–Serre Theorem, implying that the decomposition of prime numbers in a certain extension of the rationals is governed by the coefficients of the eta product \(\eta^{2}(z)\).     

On High G k−1,k -Defective Varieties

We give a rough characterization for n-dimensional varieties with G_k?1,k-defect equal to a > 0 if k ≥ n. Then we apply this in the case that a ≥ n ? 2 to become a fine classification.     

Hannover Morbiditäts- und Mortalitäts-Pflegestudie (HMMPS): Langzeitverläufe, Pflegestufenübergänge und Reaktivierungen in der gesetzlichen Pflegeversicherung

After determining the initial care level corresponding to the regulations of the German Institutional Long-Term Care Insurance, the care level may change over the course of time. Care assistance may even be cancelled owing to a reduction in necessity for care or due to the death of the recipient. From a cohort of 88,575 long term care patients of the statutory health insurance ‘‘Deutsche BKK’’ this report describes the probabilities for a change or loss of care level and death with regard to age, gender and the length of care in each level. This study thus provides a basis for actuarial calculations of long-term care insurance based on empirical data.     

The use of vector-valued martingales in risk theory

A completely dependent risk process with perturbation and phase-type distributed claim sizes is analyzed. Claim arrivals are modeled by a Markovian arrival process. Using a vector-valued martingale, the Laplace transform of the time to ruin is derived algorithmically. The conditional memoryless property of the phase-type distribution yields the distribution of the deficit at ruin as a corollary.     

Hyers–Ulam Stability of General Jensen-Type Mappings in Banach Algebras

Using the fixed point method, we prove the Hyers–Ulam stability of homomorphisms in complex Banach algebras and complex Banach Lie algebras and also of derivations on complex Banach algebras and complex Banach Lie algebras for the general Jensen-type functional equation f(α x + β y) + f(α x ? β y) = 2α f(x) for any \({\alpha, \beta \in \mathbb{R}}\) with \({\alpha, \beta \neq 0}\) . Furthermore, we prove the hyperstability of homomorphisms in complex Banach algebras for the above functional equation with α = β = 1.     

Proper restriction semigroups – semidirect products and W-products

Fountain and Gomes [4] have shown that any proper left ample semigroup embeds into a so-called W-product, which is a subsemigroup of a reverse semidirect product ${T\ltimes {\mathcal {Y}}}$ of a semilattice ${\mathcal {Y}}$ by a monoid T, where the action of T on  ${\mathcal {Y}}$ is injective with images of the action being order ideals of  ${\mathcal {Y}}$ . Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim of this paper is to give necessary and sufficient conditions on a proper left restriction semigroup such that it embeds into a W-product. We also examine the complex relationship between W-products and semidirect products of the form ${{\mathcal {Y}}\rtimes T}$ .     

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.     

Constraining by a family of strictly nonexpansive idempotent functions with applications in image reconstruction

When solving an image reconstruction problem a previous knowledge concerning the original image may lead to various constraining strategies. A convergence result has been previously proved for a constrained version of the Kaczmarz projection algorithm with a single strictly nonexpansive idempotent function with a closed image. In this paper we consider a more general projection based iterative method and a family of such constraining functions with some additional hypotheses in order to better use the a priori information for every approximation calculated. We present a particular family of box-constraining functions which satisfies our assumptions and we design an adaptive algorithm that uses an iteration-dependent family of constraining functions for some numerical experiments of image reconstruction on Tomographic Particle Image Velocimetry.     

A note on subdifferentials of pointwise suprema

The paper suggests a new approach to calculation of subdifferentials of suprema of convex functions without any qualification conditions which essentially relies on the Hirriart-Urruty–Phelps formula for subdifferentials of sums of convex l.s.c. functions (also supplied with a simple new proof). The approach in particular provides for a simpler way to (a certain generalization of) the most recent and so far most general formulas of Hantoute–López–Zalinescu and López–Volle.     

Separative Ideals,Clean Elements,and Unit-Regularity

ABSTRACT

We prove that an ideal I of a regular ring R is separative if and only if each a ? R satisfying Rr(a)aR = Ra?(a)R = RaR(1 ? a)R ? I is unit-regular. If I is a separative ideal of a regular ring R, then each a ? R satisfying Rar(a²) = ?(a²)aR = R(a ? a²) R ? I is clean. Some applications are also obtained.     

Illiquid financial market models and absence of arbitrage

We study financial market models with different liquidity effects. In the first part of this paper, we extend the short-term price impact model introduced by Rogers and Singh (2007) to a general semimartingale setup. We show the convergence of the discrete-time into the continuous-time modeling framework when trading times approach each other. In the second part, arbitrage opportunities in illiquid economies are considered, in particular a modification of the feedback effect model of Bank and Baum (2004). We demonstrate that a large trader cannot create wealth at no risk within this framework. Here we have to assume that the price process is described by a continuous semimartingale.     

The general non-symmetric,unbalanced star circuit

We provide the general solution of problems concerning AC star circuits by turning them into geometric problems. We show that one problem is strongly related to the Fermat-point of a triangle. We present a solution that is well adapted to the practical application the problem is based on. Furthermore, we solve a generalization of the geometric situation and discuss the relation to non-symmetric, unbalanced AC star circuits.     

On the efficiency of the Asmussen–Kroese-estimator and its application to stop-loss transforms

Recently, Asmussen and Kroese [6] proposed efficient stochastic estimators to approximate the probability that a random sum exceeds a certain threshold. This article aims to extend the analysis of some of these estimators: Several questions induced by [6] for the threshold setting are discussed. In particular, it is shown that Asmussen and Kroese provide the first heavy-tailed example with vanishing relative error. Furthermore, estimators of similar type for the approximation of moments of stop-loss transforms are given. Their asymptotic performance is analysed and a numerical illustration is attached at the end.     

On the continuity of superposition operators in the space of functions of bounded variation

In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014).