度量空间中的Lipschitz道路及其*积运算

通过距离空间(X,d)上Lipschitz映射,引入了X中的Lipschitz道路的概念,研究了它的一些性质,证明了同一度量空间中两个相交的Lipschitz道路的并也是Lipschitz道路。引入了度量空间X,Y中的Lipschitz道路E,F关于h:[0,1]→E与g:[0,1]→F的*-积E*F│(h,g),证明了当h,g都是Lipschitz满射时E*F│(h,g)为乘积空间X×Y中的Lipschitz道路。     

Locally lipschitz cooperative games

A locally Lipschitz cooperative generalized game is described by its coalition worth function v defined on the set [0, 1]ⁿ of generalized (or fuzzy) coalitions of n players. We assume that v is positively homogeneous and locally Lipschitz. We propose the Clarke's generalized gradient ∂v(c^N) of v at the coalition c^N=(1,…,1) of all players as a set of solutions, and we study its property. We point out that it coincides with the core when v is super–additive and to the Shapley value when v is smooth.     

On Lipschitz continuity of the iterated function system in a stochastic optimal growth model

This paper provides qualitative properties of the iterated function system (IFS) generated by the optimal policy function for a class of stochastic one-sector optimal growth models. We obtain, explicitly in terms of the primitives of the model (i) a compact interval (not including the zero stock) in which the support of the invariant distribution of output must lie, and (ii) a Lipschitz property of the iterated function system on this interval. As applications, we are able to present parameter configurations under which (a) the support of the invariant distribution of the IFS is a generalized Cantor set, and (b) the invariant distribution is singular.     

Subdifferentiability and lipschitz continuity in experimental design problems

The calculus of concave functions is a widely accepted tool for optimum experimental design problems. However, as a function of the support points and the weights the design problem fails to be concave. In this paper we make use of generalized gradients in the sense of Rockafellar (1980) and Clarke (1983). A chain rule is presented for the subdifferential of the composition of an information function with the moment matrix mapping. Lipschitz continuity of the global design function is proved and conditions for strict differentiability are given.     

关于自相似空间的一些讨论

本文通过定义距离空间(X,d)上的自相似空间的概念,给出相关命题的同时证E(kk=1,2,…,m)多于两个点的紧致距离空间(X,d)中的压缩Lipschitz道路,且X=km=1Ek。如果存在非扩张Lipschitz满射gi:X→[0,1(]1燮i燮m),则X为自相似空间;作为应用得知:有限个有公共交点的Lipschitz道路之并是自相似空间;平面上任一连通图与树都是自相似空间。     

由分数布朗运动和Poisson过程驱动的随机微分方程解的存在性和唯一性

本文主要讨论了由分数布朗运动和Poisson过程驱动的随机微分方程。当方程的系数满足Lipschitz条件和线性增长条件时,给出方程解的存在性和唯一性定理。     

Robustness for path-dependent volatility models

In this paper, we consider a generalisation of the Hobson–Rogers model proposed by Foschi and Pascucci (Decis Eocon Finance 31(1):1–20, 2008) for financial markets where the evolution of the prices of the assets depends not only on the current value but also on past values. Using differentiability of stochastic processes with respect to the initial condition, we analyse the robustness of such a model with respect to the so-called offset function, which generally depends on the entire past of the risky asset and is thus not fully observable. In doing this, we extend previous results of Blaka Hallulli and Vargiolu (2007) to contingent claims, which are globally Lipschitz with respect to the price of the underlying asset, and we improve the dependence of the necessary observation window on the maturity of the contingent claim, which now becomes of linear type, while in Blaka Hallulli and Vargiolu (2007), it was quadratic. Finally, in this framework, we give a characterisation of the stationarity assumption used in Blaka Hallulli and Vargiolu (2007), and prove that this model is stationary if and only if it is reduced to the original Hobson–Rogers model. We conclude by calibrating the model to the prices of two indexes using two different volatility shapes.     

A fixed point algorithm with economic applications

A new fixed point algorithm applicable to large scale economic problems is developed. Motivated by successive approximation the algorithm iterates on a convex combination of the traditional price to price map and the identity map. The actual combination used is determined from the convergence error of the previous iteration. The specified convex combination is guaranteed to be a contraction if the original map is Lipschitz continuous and antitonal. These sufficient conditions are shown to be weaker than the contraction requirement. A simple supply-demand example is presented to show the advantages of this algorithm over successive approximation.     

Convergence of stochastic approximation algorithms under irregular conditions

We consider a class of stochastic approximation (SA) algorithms for solving a system of estimating equations. The standard condition for the convergence of the SA algorithms is that the estimating functions are locally Lipschitz continuous. Here, we show that this condition can be relaxed to the extent that the estimating functions are bounded and continuous almost everywhere. As a consequence, the use of the SA algorithm can be extended to some problems with irregular estimating functions. Our theoretical results are illustrated by solving an estimation problem for exponential power mixture models.     

Optimal currency basket in a world of generalized floating: An application to the Nordic countries

This paper derives optimal weights for a currency basket taking into consideration the objective of policymakers in the Nordic countries. The analysis is based on the work of Branson and Katseli and Lipschitz and Sundararajan. This paper derives both export share weights by using a simple multi-country model and basket weights by assuming that the objective of the policymakers is to minimize fluctuations in the production of exports. The results show that only under special circumstances are the two weights the same. The basket weights tend to be functions of export weights and other factors such as the covariances of relative prices and exchange rate. Using the formulas derived in the paper, various optimal basket calculations are made for Norway, Finland, and Sweden.     

Consumption optimization for recursive utility in a jump-diffusion model

In this paper, we consider a market model with prices and consumption following a jump-diffusion dynamics. In this setting, we first characterize the optimal consumption plan for an investor with recursive stochastic differential utility on the basis of his/her own beliefs, then we solve the inverse problem to find what beliefs make a given consumption plan optimal. The problem is viewed in general for a class of homogeneous recursive utility, and later we choose a logarithmic model for the utility aggregator as an explicitly computable example. When beliefs, represented via Girsanov’s theorem, get incorporated into the model, the change of measure gives rise, up to a transformation, to a backward stochastic differential equation whose generator exhibits a quadratic behavior in the Brownian component and a locally Lipschitz one in the jump component, which is solvable on the basis of some recent results.     

Identification and Estimation Issues in Exponential Smooth Transition Autoregressive Models

Exponential smooth transition autoregressive (ESTAR) models are widely used in the international finance literature, particularly for the modelling of real exchange rates. We show that the exponential function is ill‐suited as a regime weighting function because of two undesirable properties. Firstly, it can be well approximated by a quadratic function in the threshold variable whenever the transition function parameter γ, which governs the shape of the function, is ‘small’. This leads to an identification problem with respect to the transition function parameter and the slope vector, as both enter as a product into the conditional mean of the model. Secondly, the exponential regime weighting function can behave like an indicator function (or dummy variable) for very large values of γ. This has the effect of ‘spuriously overfitting’ a small number of observations around the location parameter μ. We show that both of these effects lead to estimation problems in ESTAR models. We illustrate this by means of an empirical replication of a widely cited study, as well as a simulation exercise.     

Strong representation of the presmoothed quantile function estimator for censored data

We consider lifetime data subject to right random censorship. In this context, this paper deals with the topic of estimating the distribution function of the lifetime and the corresponding quantile function. As it has been shown that the classical Kaplan–Meier estimator of the distribution function can be improved by means of presmoothing ideas, we introduce a quantile function estimator via the presmoothed distribution function estimator studied by Cao et al. [Journal of Nonparametric statistics, Vol. 17 (2005) pp. 31–56.] The main result of this paper is an almost sure representation of this presmoothed estimator. As a consequence, its strong consistency and asymptotic normality are established. The performance of this new quantile estimator is analyzed in a simulation study and applied to a real data example.     

Two-stage rank estimation of quantile index models

This paper estimates a class of models which satisfy a monotonicity condition on the conditional quantile function of the response variable. This class includes as a special case the monotonic transformation model with the error term satisfying a conditional quantile restriction, thus allowing for very general forms of conditional heteroscedasticity. A two-stage approach is adopted to estimate the relevant parameters. In the first stage the conditional quantile function is estimated nonparametrically by the local polynomial estimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246–269; Annals of Statistics 19 (1991b) 760–777) and Cavanagh (1996, Preprint). In the second stage, the monotonicity of the quantile function is exploited to estimate the parameters of interest by maximizing a rank-based objective function. The proposed estimator is shown to have desirable asymptotic properties and can then also be used for dimensionality reduction or to estimate the unknown structural function in the context of a transformation model.     

Constitutional implementation

We consider the problem of implementing a social choice correspondence H in Nash equilibrium when the constitution of the society is given by an effectivity function E. It is assumed that the effectivity function of , is a sub-correspondence of E. We found necessary and efficient conditions for a game form to implement H (in Nash equilibria), and to satisfy, at the same time, that , the effectivity function of , is a sub-correspondence of (which guarantees that is compatible with E). We also find sufficient conditions for the coincidence of the set of winning coalitions of and , and for . All our results are sharp as is shown by suitable examples. Received: 15 December 2000 / Accepted: 3 September 2001     

α-optimal sampling plans for lot-by-lot defects inspection

Investigations on acceptance sampling have attached rather few importance to defects inspection. For modern quality control, both the steadily increasing complexity of products and the need for differentiated cost calculation involve a clear demand for economic defects sampling in its practically most relevant form: lot-by-lot single sampling plans, where the OC (lot OC) is considered as a function of lot sizeN, sample sizen, acceptance numberc, number of defects in the lotK. Starting from an appropriate lot OC function, the present paper develops an economic cost and control model and an economic objective function for single defects sampling plans by adapting theα-optimal sampling scheme, introduced by E. von Collani for defectives inspection, to the purposes of defects inspection. A simple and accurate approximation ofα-optimal defects plans is derived by means of a Poisson approximation of the lot OC function.     

Sequential estimation of a linear function of normal means under asymmetric loss function

The problem of estimating a linear function of k normal means with unknown variances is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, sequential stopping rules satisfying a general set of assumptions are considered. Two estimators are proposed and second-order asymptotic expansions of their risk functions are derived. It is shown that the usual estimator, namely the linear function of the sample means, is asymptotically inadmissible, being dominated by a shrinkage-type estimator. An example illustrates the use of different multistage sampling schemes and provides asymptotic expansions of the risk functions. Received: August 1999     

Total Factor Productivity Growth and Technical Change in a Profit Function Framework

This paper develops a framework for measuring and decomposing TFP changes, within the parametric approach, by using directly the estimated parameters of a profit function. Two alternative relationships are derived for measuring and decomposing TFP changes via a profit function based on two alternative definitions of the rate of technical change, i.e., input- and output-based. Initially a long-run equilibrium framework is assumed and then the analysis is extended to the case of temporary equilibrium. The latter framework is applied to US agriculture by estimating a translog profit function and analyzing TFP changes during the period 1948–1994.     

Scale Efficiency in Canadian Trust Companies

A new method for estimating a relative scale function is derived which does not require the specification or estimation of the firm's cost function. The method is applied to a panel of Canadian trust companies for the years 1985–1988. Results show that the industry scale function is convex with respect to firm size, as measured by the firm's assets. The method avoids a number of shortcomings associated with traditional studies of scale economies in financial institutions.     

独立学院数学改革之数学实验——Matlab平台上“四步法”研究函数的单调性与凹凸性

数学实验是高等数学改革的重要方向,借助数学软件简化数学计算,注重数学应用是数学教学的一个新动向。函数的单调性与凹凸性是高等数学中导数应用部分的一个重要内容。本文借助功能强大的数学软件Matlab,巧用计算与图形功能,提出应用导数研究函数的单调性与凹凸性的四步法,即(1)求导函数;(2)求导函数的零点;(3)画原函数与导函数图;(4)确定函数的单调性或凹凸性。该方法采用先求导函数零点再画图的顺序,确保画图区域包含导函数的零点,避免遗漏极值点或拐点,进一步通过实例系统体现该方法对函数单调性与凹凸性的可视化判定。