Weakly Convex Sets and Their Properties

In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.     

具有时滞响应的择好期权定价

假设市场是完备的,在文中使用了计价单位变换,等价鞅测度理论和无套利原理研究了股票价格具有时滞的欧式择好期权,得到了欧式择好期权的定价公式和对冲交易策略。     

Modeling Capital Gains Taxes for Trading Strategies of Infinite Variation

In this article, we show that the payment flow of a linear tax on trading gains from a security with a semimartingale price process can be constructed for all càglàd and adapted trading strategies. It is characterized as the unique continuous extension of the tax payments for elementary strategies w.r.t. the convergence “uniformly in probability.” In this framework, we prove that under quite mild assumptions dividend payoffs have almost surely a negative effect on investor’s after-tax wealth if the riskless interest rate is always positive. In addition, we give an example for tax-efficient strategies for which the tax payment flow can be computed explicitly.     

On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1

The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using the Pontryagin maximum principle, we treat Riemannian and sub-Riemannian cases in a unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way, first we obtain a new elementary proof of the classical Levi-Civita theorem on the classification of all Riemannian geodesically equivalent metrics in a neighborhood of the so-called regular (stable) point w.r.t. these metrics. Second, we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally, we give a classification of all pairs of geodesically equivalent Riemannian metrics on a surface that are proportional at an isolated point. This is the simplest case, which was not covered by Levi-Civita’s theorem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

Qualitative and infinitesimal robustness of tail-dependent statistical functionals

The main goal of this article is to introduce a new notion of qualitative robustness that applies also to tail-dependent statistical functionals and that allows us to compare statistical functionals in regards to their degree of robustness. By means of new versions of the celebrated Hampel theorem, we show that this degree of robustness can be characterized in terms of certain continuity properties of the statistical functional. The proofs of these results rely on strong uniform Glivenko-Cantelli theorems in fine topologies, which are of independent interest. We also investigate the sensitivity of tail-dependent statistical functionals w.r.t. infinitesimal contaminations, and we introduce a new notion of infinitesimal robustness. The theoretical results are illustrated by means of several examples including general L- and V-functionals.     

On stability and Bohl exponent of linear singular systems of difference equations with variable coefficients

In this paper we investigate the stability of linear singular systems of difference equations with variable coefficients by the projector-based approach. We study the preservation of uniform/exponential stability when the system coefficients are subject to allowable perturbations. A Bohl–Perron type theorem is obtained which provides a necessary and sufficient condition for the boundedness of solutions of nonhomogenous systems. The notion of Bohl exponent is introduced and we characterize the relation between the exponential stability and the Bohl exponent. Finally, robustness of the Bohl exponent with respect to allowable perturbations is investigated.     

Baskakov型算子同时逼近的点态估计

In this paper, for Baskakov, Baskakov-Kantorovich and Baskakov-Durrmyer operators Ln(f,x),we give a simultaneous approximation of equivalent theorem with ω^2ψλ (f, t) The theorem unites the corrosponding results of classical and the Ditzian-Totik moduli of smoothness.     

Stochastic scheduling problems II-set strategies-

The paper introduces the finite class of set strategies for stochastic scheduling problems. It is shown that the knownstable classes of strategies such as ES and MES strategies are of this type, as arelist-scheduling strategies such as LEPT and SEPT and other, more complicatedpriority-type strategies. Roughly speaking, set strategies are characterized by the fact that the decision as to which jobs should be started at timet depends only on the knowledge of the two sets of jobs finished up to timet and being processed at timet. Contrary to list scheduling strategies, set strategies may involve deliberate idleness of machines, i.e. may not be greedy and can therefore not generally be induced by priority rules. It is demonstrated that set strategies have useful properties. They are e.g. ⁿ -almost everywhere continuous and therefore show satisfactorystability behaviour w.r.t. weak convergence of the joint distribution of job durations. Furthermore, the optimum w.r.t.all strategies is already attained on this class if job durations are independent and exponentially distributed and the performance measure fulfills a certainshift condition. This shift property is a quite natural concept and generalizes aspects of the notion ofadditivity in semi-Markov decision theory and stochastic dynamic optimization. Its complete analytical characterization is a major object of this paper. Typical additive cost criteria such as makespan and flowtime are of course covered, which yields simultaneously a first step towards generalization of optimality of LEPT and SEPT rules, as known for special cases. In fact, in view of the obtained optimality result, the question of when deliberate idleness of machines can be avoided, gains considerable interest, as it characterizes stochastic environments in whichpriority strategies are optimal. This provides a major link with current research on the analysis of networks of queues in the context of computer systems.The work of the first two authors was supported by the Minister für Wissenschaft und Forschung des Landes Nordrhein-Westfalen, while the work of the last author was supported by DAAD.     

Estimating the efficient price from the order flow: A Brownian Cox process approach

At the ultra high frequency level, the notion of price of an asset is very ambiguous. Indeed, many different prices can be defined (last traded price, best bid price, mid price, etc.). Thus, in practice, market participants face the problem of choosing a price when implementing their strategies. In this work, we propose a notion of efficient price which seems relevant in practice. Furthermore, we provide a statistical methodology enabling to estimate this price from the order flow.     

Paths,Homotopy and Reduction in Digital Images

The development of digital imaging (and its subsequent applications) has led to consideration and investigation of topological notions that are well-defined in continuous spaces, but not necessarily in discrete/digital ones. In this article, we focus on the classical notion of path. We establish in particular that the standard definition of path in algebraic topology is coherent w.r.t. the ones (often empirically) used in digital imaging. From this statement, we retrieve, and actually extend, an important result related to homotopy-type preservation, namely the equivalence between the fundamental group of a digital space and the group induced by digital paths. Based on this sound definition of paths, we also (re)explore various (and sometimes equivalent) ways to reduce a digital image in a homotopy-type preserving fashion.     

离散时间金融市场中的增长最优投资组合

在这篇文章中我们给出了离散时间不完全金融市场中增长最优投资组合的显式表达式，然后用计价单位投资组合法给出了期权的定价。     

σ-kompakte Räume

The theorem, that -compact spaces are Lindelöf, is equivalent to the countable axiom of choice. Variants of this theorem are compared with weak versions of the axiom of choice.     

Optimal replenishment policy for the EPQ model with permissible delay in payments and allowable shortages

The main purpose of this paper is to investigate the optimal replenishment policy under conditions of permissible delay in payments and allowable shortages within the economic production quantity (EPQ) framework. We extend the work of Chung and Huang [15] to assume that the replenishment rate is finite and the unit selling price is not necessarily equal to the unit purchasing price. A theorem is developed to determine the optimal replenishment policy. Finally, numerical examples are given to illustrate the theorem.     

Coherent Homotopical Algebras: “Special Gamma-Categories”

We introduce a theory of coherence for symmetric monoidal categories inthe spirit of Segal and show that it is equivalent, in an appropriate sense,to MacLanes original notion. More precisely, we prove thatspecial categories, the analogue ofspecial spaces, and coherently symmetric monoidalcategories are one and the same. This is analogous to the situation intopology where special spaces are precisely homotopicalcommutative monoids. In light of the obervation that the category of smallcategories Cat bears a functorial Quillen model structure with respect tothe class of categorical equivalences: in fact, is a homotopy theory in thesense of Heller, we may reinterpret the theorem as stating that coherentlysymmetric monoidal categories are precisely the homotopical commutativemonoids within this new homotopy theory.     

Binary games in constitutional form and collective choice

In this paper, we define the notion of binary game in constitutional form. For this game, we define a core and give a necessary and sufficient condition for a game to be stable.We define a representation of a collective choice rule by a binary game in constitutional form and characterize those collective choice rules which are representable.We finally introduce the notion of c-social decision function and characterize, as an application of our theorem on stability of binary constitutional games, the collective choice rules which are c-social decision functions.Our representation of a collective choice rule by a binary game in constitutional form is an obvious improvement of the classical representation by a simple game.     

Asymptotic power utility-based pricing and hedging

Kramkov and Sîrbu (Ann. Appl. Probab., 16:2140–2194, 2006; Stoch. Proc. Appl., 117:1606–1620, 2017) have shown that first-order approximations of power utility-based prices and hedging strategies for a small number of claims can be computed by solving a mean-variance hedging problem under a specific equivalent martingale measure and relative to a suitable numeraire. For power utilities, we propose an alternative representation that avoids the change of numeraire. More specifically, we characterize the relevant quantities using semimartingale characteristics similarly as in ?erný and Kallsen (Ann. Probab., 35:1479–1531, 2007) for mean-variance hedging. These results are illustrated by applying them to exponential Lévy processes and stochastic volatility models of Barndorff-Nielsen and Shephard type (J. R. Stat. Soc. B, 63:167–241, 2001). We find that asymptotic utility-based hedges are virtually independent of the investor’s risk aversion. Moreover, the price adjustments compared to the Black–Scholes model turn out to be almost linear in the investor’s risk aversion, and surprisingly small unless very high levels of risk aversion are considered.     

Normal hyperimaginaries

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of the Lascar-Pillay theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of the Peter–Weyl theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types for bounded hyperimaginaries.     

基于在线评论的多个竞争制造商和零售商的产品定价策略

在线评论作为一种产品信息传播载体,越来越受到网上电商及消费者的重视,并在很大程度上影响消费者的购买决策。本文在多个竞争性制造商为在线零售商提供可替代性产品并通过零售商销售给网络消费者的电子商务环境下,研究在线评论信息如何影响网络消费者购买决策及在线零售商和制造商的定价策略。以neo-Hoteling模型为基础,构建了依赖零售渠道在线评论的消费者选择模型,并通过模型求解定量分析了二级供应链结构分散系统下在线评论对多个竞争性制造商及零售商最优决策的影响。得到当制造商基于评论制定最优定价策略时,在线评论对市场竞争强度没有影响,但决定潜在市场大小;各产品的均衡批发价及销售价按一定的比例随评论揭示的该产品与其他产品质量均值之差(正或负)增加或减少,评论信息通常会使制造商因好评而获利,由于评论增加了不同产品需求的不对称性,零售商因而具有更大的调价空间,往往通过提高(降低)占据有利(不利)评论的产品价格获得更高的利润。     

Implementation via mechanisms with transfers

We consider the following problem: suppose that the social choice rule is given, but the planner has the opportunity to redistribute among agents some numeraire commodity (“money”) by compensatory transfers; can the planner find a transfer such that the social choice rule is Nash implementable by the transfer?In this paper, we establish a necessary and sufficient condition for Nash implementation by transfers. Furthermore, we construct a simplified mechanism, which Nash-implements the desired social choice rule by transfers.