Optimal reinsurance with positively dependent risks

In the individual risk model, one is often concerned about positively dependent risks. Several notions of positive dependence have been proposed to describe such dependent risks. In this paper, we assume that the risks in the individual risk model are positively dependent through the stochastic ordering (PDS). The PDS risks include independent, comonotonic, conditionally stochastically increasing (CI) risks, and other interesting dependent risks. By proving the convolution preservation of the convex order for PDS random vectors, we show that in individualized reinsurance treaties, to minimize certain risk measures of the retained loss of an insurer, the excess-of-loss treaty is the optimal reinsurance form for an insurer with PDS dependent risks among a general class of individualized reinsurance contracts. This extends the study in Denuit and Vermandele (1998) on individualized reinsurance treaties to dependent risks. We also derive the explicit expressions for the retentions in the optimal excess-of-loss treaty in a two-line insurance business model.     

On the invariant properties of notions of positive dependence and copulas under increasing transformations

Notions of positive dependence and copulas play important roles in modeling dependent risks. The invariant properties of notions of positive dependence and copulas under increasing transformations are often used in the studies of economics, finance, insurance and many other fields. In this paper, we examine the notions of the conditionally increasing (CI), the conditionally increasing in sequence (CIS), the positive dependence through the stochastic ordering (PDS), and the positive dependence through the upper orthant ordering (PDUO). We first use counterexamples to show that the statements in Theorem 3.10.19 of Müller and Stoyan (2002) about the invariant properties of CIS and CI under increasing transformations are not true. We then prove that the invariant properties of CIS and CI hold under strictly increasing transformations. Furthermore, we give rigorous proofs for the invariant properties of PDS and PDUO under increasing transformations. These invariant properties enable us to show that a continuous random vector is PDS (PDUO) if and only of its copula is PDS (PDUO). In addition, using the properties of generalized left-continuous and right-continuous inverse functions, we give a rigorous proof for the invariant property of copulas under increasing transformations on the components of any random vector. This result generalizes Proposition 4.7.4 of Denuit et al. (2005) and Proposition 5.6. of McNeil et al. (2005).     

Permutohedra and minimal matrices

The notions of minimality, π-uniqueness and additivity originated in discrete tomography. They have applications to Kronecker products of characters of the symmetric group and arise as the optimal solutions of quadratic transportation problems. Here, we introduce the notion of real-minimality and give geometric characterizations of all these notions for a matrix A, by considering the intersection of the permutohedron determined by A with the transportation polytope in which A lies. We also study the computational complexity of deciding if the properties of being additive, real-minimal, π-unique and minimal hold for a given matrix, and show how to efficiently construct some matrix with any of these properties.     

Various notions of amenability in Banach algebras

We give a survey of results and problems concerning various notions of amenability in Banach algebras. We also provide different characterizations of these notions of amenability and the relationship that exists between them and some important properties of the algebra.     

Theory of subdualities

We present a new theory of dual systems of vector spaces that extends the existing notions of reproducing kernel Hilbert spaces and Hilbert subspaces. In this theory, kernels (understood as operators rather than kernel functions) need not be positive or self-adjoint. These dual systems, called subdualities, enjoy many properties similar to those of Hilbert subspaces and include the notions of Hilbert subspaces or Kreîn subspaces as particular cases. Some applications to Green operators or invariant subspaces are given.     

Asset proportions in optimal portfolios with dependent default risks

In this note, we consider the dependent default risk model of factor type. The dependence between the returns of assets is driven by default indicators. Sufficient conditions on the dependence structure of default indicators and on the utility function are investigated which enable one to order the optimal amount invested in each asset. We thus complement one result in [Cheung, K.C., Yang, H., 2004. Ordering optimal proportions in the asset allocation problem with dependent default risks. Insurance: Math. Econom. 35, 595-609].     

Leibniz interpolation properties

We introduce a family of notions of interpolation for sentential logics. These concepts generalize the ones for substructural logics introduced in [5]. We show algebraic characterizations of these notions for the case of equivalential logics and study the relation between them and the usual concepts of Deductive, Robinson, and Maehara interpolation properties.     

Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency

This article aims to elaborate on various notions of Levitin–Polyak well-posedness for set optimization problems concerning Pareto efficient solutions. These notions are categorized into two classes including pointwise and global Levitin–Polyak well-posedness. We give various characterizations of both pointwise and global Levitin–Polyak well-posedness notions for set optimization problems. The hierarchical structure of their relationships is also established. Under suitable conditions on the input data of set optimization problems, we investigate the closedness of Pareto efficient solution sets in which they are different from the weakly efficient ones. Furthermore, we provide sufficient conditions for global Levitin–Polyak well-posedness properties of the reference problems without imposing the information on efficient solution sets.

     

Characterization of dependence concepts in normal distributions

Summary In the present paper we deal with the characterization of some dependence concepts for the multivariate normal distribution. It turns out that normal distributions have some special properties w.r.t. these dependence concepts and, furthermore, that the characterizations are closely connected to some interesting problems on matrices. Some applications to simultaneous confidence bounds are discussed.     

Concepts of curvatures in normed planes

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.     

Nested Conditional Value-at-Risk portfolio selection: A model with temporal dependence driven by market-index volatility

In a multistage stochastic programming framework, we develop a new method for finding an approximated portfolio allocation solution to the nested Conditional Value-at-Risk model when asset log returns are stagewise dependent. We describe asset log returns through a single-factor model where the driving factor is the market-index log return modeled by a Generalized Autoregressive Conditional Heteroskedasticity process to take into account the serial dependence usually observed. To solve the nested Conditional Value-at-Risk model, we implement a backward induction scheme coupled with cubic spline interpolation that reduces the computational complexity of the optimal portfolio allocation and allows to treat problems otherwise unmanageable.     

Levitin–Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints

In this paper we consider strong bilevel vector equilibrium problems and introduce the concepts of Levitin–Polyak well-posedness and Levitin–Polyak well-posedness in the generalized sense for such problems. The notions of upper/lower semicontinuity involving variable cones for vector-valued mappings and their properties are proposed and studied. Using these generalized semicontinuity notions, we investigate sufficient and/or necessary conditions of the Levitin–Polyak well-posedness for the reference problems. Some metric characterizations of these Levitin–Polyak well-posedness concepts in the behavior of approximate solution sets are also discussed. As an application, we consider the special case of traffic network problems with equilibrium constraints.     

Strong Abadie CQ,ACQ, calmness and linear regularity

The Abadie CQ (ACQ) for convex inequality systems is a fundamental notion in optimization and approximation theory. In terms of the contingent cone and tangent derivative, we extend the Abadie CQ to more general convex multifunction cases and introduce the strong ACQ for both multifunctions and inequality systems. Some seemly unrelated notions are unified by the new ACQ and strong ACQ. Relationships among ACQ, strong ACQ, basic constraint qualification (BCQ) and strong BCQ are discussed. Using the strong ACQ, we study calmness of a closed and convex multifunction between two Banach spaces and, different from many existing dual conditions for calmness, establish several primal characterizations of calmness. As applications, some primal characterizations for error bounds and linear regularity are established; in particular, some existing results are improved.     

On Searching for Large Objects with Small Probes: A Search Model for Exploration

We propose a model for optimal allocation of effort when searching for large objects with small probes. Because of the size of the objects, search activities at different locations can be physically and statistically dependent, and consequently classical search models are inappropriate. The difficulties resulting from such statistical dependence can be overcome by using original data or Monte-Carlo derived scenarios in an integer linear programming model. The proposed optimization model should be of moderate size in typical applications, and by exploiting its special structure it can be solved optimally by standard methods. When this is uneconomical it may be solved approximately by efficient heuristics. Applications of these ideas have already been made to the detection of glaucoma. This paper focuses on possible applications in exploratory drilling for petroleum.     

On allocations to portfolios of assets with statistically dependent potential risk returns

This note studies how the allocation impacts on the expected potential return of the portfolio of risk assets with some new dependence structures characterized through the orthant probability of their potential returns. As applications, we revisit the financial risk model and actuarial default risk model, and study the dependence structure of potential risk returns and the utility functions such that in the optimal allocations the assets are arranged in ascending order. The main results complement some related ones of Cheung and Yang (2004) and Chen and Hu (2008).     

Small embedding characterizations for large cardinals

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß.     

Structural properties of the optimal resource allocation policy for single-queue systems

This paper studies structural properties of the optimal resource allocation policy for single-queue systems. Jobs arrive at a service facility and are sent one by one to a pool of computing resources for parallel processing. The facility poses a constraint on the maximum expected sojourn time of a job. A central decision maker allocates the servers dynamically to the facility. We consider two models: a limited resource allocation model, where the allocation of resources can only be changed at the start of a new service, and a fully flexible allocation model, where the allocation of resources can also change during a service period. In these two models, the objective is to minimize the average utilization costs whilst satisfying the time constraint. To this end, we cast these optimization problems as Markov decision problems and derive structural properties of the relative value function. We show via dynamic programming that (1) the optimal allocation policy has a work-conservation property, and (2) the optimal number of servers follows a step function with as extreme policy the bang-bang control policy. Moreover, (3) we provide conditions under which the bang-bang control policy takes place. These properties give a full characterization of the optimal policy, which are illustrated by numerical experiments.     

Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets

Motivated by the subsmoothness of a closed set introduced by Aussel et al. (2005) [8], we introduce and study the uniform subsmoothness of a collection of infinitely many closed subsets in a Banach space. Under the uniform subsmoothness assumption, we provide an interesting subdifferential formula on distance functions and consider uniform metric regularity for a kind of multifunctions frequently appearing in optimization and variational analysis. Different from the existing works, without the restriction of convexity, we consider several fundamental notions in optimization such as the linear regularity, CHIP, strong CHIP and property (G) for a collection of infinitely many closed sets. We establish relationships among these fundamental notions for an arbitrary collection of uniformly subsmooth closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of closed convex sets to the nonconvex setting.     

Set optimization by means of variable order relations

We introduce several variable order relations to compare sets in a linear topological space and we consider set optimization problems equipped with these variable ordering structures. By considering a set approach, we introduce characterizations for optimal solutions and we provide a kind of vectorization result to obtain solutions of set optimization problems equipped with the introduced variable order relations.     

On extensions of the core and the anticore of transferable utility games

We consider several related set extensions of the core and the anticore of games with transferable utility. An efficient allocation is undominated if it cannot be improved, in a specific way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undominated set, and we show that it consists of finitely many polytopes with a core-like structure. One of these polytopes is the $L_1$ -center, consisting of all efficient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The $L_1$ -center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a single-valued selection from the excess Pareto optimal set, the min-prenucleolus, which is defined as the prenucleolus of the minimum of a game and its dual.