Ein zweidimensionales Durationskonzept

The German proposal for a Solvency II-compatible standard model for the life insurance branch calculates the risk capital that is necessary for a sufficient risk capitalisation of the company at hand. This capital is called ‘‘target capital’’ or Solvency Capital Requirement (SCR for short). For this to achieve it is applied the book value of the actuarial reserve onto the well-known market value formula getting the market value (or present value) by means of the classical duration concept as a global approach (cf. the documentation of the standard model of the GDV p. 26). This formula takes into account the impact of the interest rate but leaves aside all the other actuarial assumptions. In particular, the influence of the biometrical assumptions is not considered. This is at least one reason, why this ansatz is – at the time being – no more compatible with the Solvency II requirements and thus does no more satisfy its own entitlements. In the work at hand it is proposed and worked out a concept that overcomes this drawback. The result is a formula with the help of which the present value of the actuarial liabilities is calculated from their book value in fact by taking into account the interest rate as well as the biometrical assumptions. It is to be remarked that the proposed two-dimensional duration concept may be developed completely along the lines given by the classical one-dimensional analogue leaving some arbitraries only on determining the biometrical gauge, i.e., the mapping of the vector that represents the formula of the active lives remaining onto its average value. For this to achieve one has to consider the underlying business in force. The superordinate relevance of such a two-dimensional ansatz lies in the fact that the developments of the project Solvency II during the last months have shown that its success depends crucially on the availability of efficient and well-elaborated approximation procedures.     

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 \(\eta^{2}(z)\).     

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.     

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.     

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 ${{\mathcal {Y}}\rtimes T}$ .     

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.     

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.     

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.     

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.     

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.     

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.     

Finite element convergence for the Joule heating problem with mixed boundary conditions

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes.     

Approximate homomorphisms on lattices

We prove two results concerning an Ulam-type stability problem for homomorphisms between lattices. One of them involves estimates by quite general error functions; the other deals with approximate (join) homomorphisms in terms of certain systems of lattice neighborhoods. As a corollary, we obtain a stability result for approximately monotone functions.     

Landesman–Lazer conditions for difference equations involving sublinear perturbations

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.     

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 spaces with a countable μ-base

In topological spaces every normal space with a countable base is metacompact. We show that this is not necessarily true in generalized topological spaces; more exactly we give an example of a μ-normal space with a countable μ-base which has a μ-open cover with no μ-open point-finite refinement.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

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.