On Jensen’s inequality and Hölder’s inequality for g-expectation

In this paper, we shall prove that for n > 1, the n-dimensional Jensen inequality holds for the g-expectation if and only if g is independent of y and linear with respect to z, in other words, the corresponding g-expectation must be linear. A Similar result also holds for the general nonlinear expectation defined in Coquet et al. (Prob. Theory Relat. Fields 123 (2002), 1–27 or Peng (Stochastic Methods in Finance Lectures, LNM 1856, 143–217, Springer-Verlag, Berlin, 2004). As an application of a special n-dimensional Jensen inequality for g-expectation, we give a sufficient condition for g under which the Hölder’s inequality and Minkowski’s inequality for the corresponding g-expectation hold true.     

Jensen's inequality for filtration consistent nonlinear expectation without domination condition

In this paper, the general filtration consistent nonlinear expectation defined on the integrable variable space is considered, based on the results in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27]. Under a natural continuous assumption for the nonlinear expectation, which weakens the domination assumption in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27], the author obtains the necessary and sufficient conditions under which Jensen's inequality for filtration consistent nonlinear expectation holds in general, respectively on scalar function and bivariate function. These two results generalize the known results on Jensen's inequality for g-expectation in [Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 1, C. R. Acad. Sci. Paris Ser. I 337 (11) (2003) 725-730; Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 2, C. R. Acad. Sci. Paris Ser. I 337 (12) (2003) 797-800; L. Jiang, On Jensen's inequality of bivariate function for g-expectation, J. Shandong Univ. 38 (5) (2003) 13-22 (in Chinese); L. Jiang, Z. Chen, On Jensen's inequality for g-expectation, Chinese Ann. Math. Ser. B 25 (3) (2004) 401-412; L. Jiang, Jensen's inequality for backward stochastic differential equation, Chinese Ann. Math. Ser. B 27 (5) (2006) 553-564; S. Fan, Jensen's inequality for g-expectation on convex (concave) function, Chinese Ann. Math. Ser. A 27 (5) (2006) 635-644 (in Chinese)].     

Hyperfinite construction of G-expectation

The hyperfinite G-expectation is a nonstandard discrete analogue of G-expectation (in the sense of Robinsonian nonstandard analysis). A lifting of a continuous-time G-expectation operator is defined as a hyperfinite G-expectation which is infinitely close, in the sense of nonstandard topology, to the continuous-time G-expectation. We develop the basic theory for hyperfinite G-expectations and prove an existence theorem for liftings of (continuous-time) G-expectation. For the proof of the lifting theorem, we use a new discretization theorem for the G-expectation (also established in this paper, based on the work of Dolinsky et al. [Weak approximation of G-expectations, Stoch. Process. Appl. 122(2) (2012), pp. 664–675]).     

Area orthogonality in normed linear spaces

A new concept of orthogonality in real normed linear spaces is introduced. Typical properties of orthogonality (homogeneity, symmetry, additivity, ...) and relations between this orthogonality and other known orthogonalities (Birkhoff, Boussouis, Unitary-Boussouis and Diminnie) are studied. In particular, some characterizations of inner product spaces are obtained.     

Lévy过程驱动的BSDE的反比较定理与Jensen不等式

本文研究了由一维Lévy过程驱动的倒向随机微分方程(BSDE)的反比较定理.利用一般g-期望下BSDE的反比较定理的证明方法,推导出了一般f-期望下BSDE的反比较定理,并给出了一般f-期望下Jensen不等式成立的充分必要条件.     

Itô’s calculus under sublinear expectations via regularity of PDEs and rough paths

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the $G$-expectation theory.     

Constructing sublinear expectations on path space

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G$G$-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G$G$-expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.     

条件g-期望的矩不等式

找到了几个使条件g-期望的矩不等式在一般意义下成立的关于g和g-期望的充分条件.     

数控车床可靠性分配中的单准则排序方法

数控车床可靠性分配模型是一个层次结构,可靠性分配的关键技术是确定结构底层指标关于顶层目标的重要度排序,其前提条件是单准则排序已知.AHP通过构造"两两比较"的"1~9"比例标度判断矩阵A_n为单准则排序提供了合理的数据条件;但是基于A_n一致性检验的特征根排序法因临界值的确定缺乏必要理论基础而受到质疑.改进AHP的FAHP因为没有摆脱"一致性检验"的干扰所以改进并不成功.为了建立与"一致性"无关的单准则排序方法定义具有可加性的评分标度概念,通过标度转换将比例标度判断矩阵A_n转化为评分标度判断矩阵B_n,利用评分标度的可加性在准则C下对n个比较对象排序.因为B_n不是正矩阵所以不存在"一致性概念",因此基于评分标度判断矩阵的排序与"一致性"无关.     

Utility maximization under g*-expectation

In this article, we introduce a nonlinear expectation, called g*-expectation, based on g-expectation and consider the optimal utility under g*-expectation in the market with a risk-free bond and d risky stocks in finite trading interval [0, T]. We construct a stochastic family by taking advantage of the comparison theorem of backward stochastic differential equations and the g*-martingale. We generalize the results of Hu et al. (Annals of Applied Probability 28(2):1691–1712, 2005), and obtain the explicit forms of the optimal trading strategies both for exp?-utility and the power utility, when g(t, z) = β_t|z|² + γ_tz.     

On positive homogeneity and comonotonic additivity of the principle of equivalent utility under Cumulative Prospect Theory

Investigating the principle of equivalent utility under Cumulative Prospect Theory, Kałuszka and Krzeszowiec (2012) established characterizations of several important properties of the premium. It turns out that the results concerning positive homogeneity and comonotonic additivity are in general not true. The aim of this paper is to present modified and essentially generalized versions of the mentioned above results.     

Strict comparison theorems under sublinear expectations

In this paper, we consider strict comparison theorems in the framework of G-expectation, which is a type of sublinear expectation associated with fully nonlinear parabolic partial differential equations. In particular, we first apply Krylov–Safonov estimates to establish the strict comparison theorem for functions from the Lipschitz class \(Lip(\Omega )\). Then we prove generalized strict comparison theorems on the enlarged space \(L_G^1(\Omega )\), which is the Banach completion of \(Lip(\Omega )\) under the G-expectation.     

非线性纵向数据模型中方差和自相关系数的齐性检验

刻画纵向数据的协方差结构有三个可能因素：随机效应、序列相关和随机误差．在纵向数据分析中，模型方差的齐性是一个基本假定．但是，该假设未必正确．Zhang和、Weiss^[1]研究了具有随机效应的线性模型的异方差检验．林金官和韦博成^[2]将Zhang和、Weiss^[1]的结果推广到非线性情形．本文对具有自相关误差的非线性纵向数据模型，研究了方差齐性和相关系数的齐性检验，得到了检验的score统计量并应用于血浆渗透数据(见Davidian和Giltian^[3])．最后，本文还给出了模拟结果．     

Background harmonic superfields inN=2 supergravity

A modification of the harmonic superfield formalism in D=4, N=2 supergravity is considered using an additional condition of covariance under the background supersymmetry with a central charge (B-covariance). Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 116, No. 2, pp. 288–304, August. 1998.     

On Jensen’s inequality for g-expectation and for nonlinear expectation

In this paper, we give a necessary and sufficient condition for g under which Jensen’s inequality holds for g-expectation. In particular, we show that if Jensen’s inequality holds for g-expectation, then g is independent of y and g is superhomogeneous. We also establish a necessary and sufficient condition under which Jensen’s inequality holds for a general filtration-consistent nonlinear expectation. Received: 18 January 2005     

Predictive Inference for the Elliptical Linear Model

This paper derives the prediction distribution of future responses from the linear model with errors having an elliptical distribution with known covariance parameters. For unknown covariance parameters, the marginal likelihood function of the parameters has been obtained and the prediction distribution has been modified by replacing the covariance parameters by their estimates obtained from the marginal likelihood function. It is observed that the prediction distribution with elliptical error has a multivariate Student'st-distribution with appropriate degrees of freedom. The results for some special cases such as the Intra-class correlation model, AR(1), MA(1), and ARMA(1,1) models have been obtained from the general results. As an application, theβ-expectation tolerance region has been constructed. An example has been added.     

Nonadditive Set Functions on a Finite Set and Linear Inequalities

A set function is a function whose domain is the power set of a set, which is assumed to be finite in this paper. We treat a possibly nonadditive set function, i.e., a set function which does not satisfy necessarily additivity, ?(A) + ?(B) = ?(A ∪ B) forA ∩ B = ∅, as an element of the linear space on the power set. Then some of the famous classes of set functions are polyhedral in that linear space, i.e., expressed by a finite number of linear inequalities. We specify the sets of the coefficients of the linear inequalities for some classes of set functions. Then we consider the following three problems: (a) the domain extension problem for nonadditive set functions, (b) the sandwich problem for nonadditive set functions, and (c) the representation problem of a binary relation by a nonadditive set function, i.e., the problem of nonadditive comparative probabilities.     

Extension and Application of Itô's Formula Under G-Framework

In continuing his study of the intrinsically nonlinear expectation and conditional expectation under the so-called G-framework, Peng introduced a nonlinear Itô calculus; here, the G refers to the generator of a nonlinear heat equation. There, he derived the corresponding Itô formula for C ²-functions with bounded Lipschiz derivatives. This restrictive class of functions limits its applicatory value to stochastic finances and cannot be applied to study the powers of the G-Brownian motion. We extend the Itô formula to a slightly more general class of functions (C ²-functions with uniformly continuous derivatives). This enables us to compute the G-expectations of the even powers of the G-Brownian motion. The G-expectation of odd powers behave differently; in particular, we show that the G-expectation of the cube of the G-Brownian motion is positive, which is qualitatively different from the classical Brownian motion case. We remark that we are not able to get a formula for the G-expectation of the general odd powers of the G-Brownian motion.     

A contribution to the linear programming approach to joint cost allocation: Methodology and application

The linear programming (LP) approach has been commonly proposed for joint cost allocation purposes. Within a LP framework, the allocation rules are based on a marginal analysis. Unfortunately, the additivity property which is required to completely allocate joint costs fails in presence of capacity, institutional or environmental constraints.     

On cardinal characteristics of Yorioka ideals

Yorioka introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in . We construct a matrix iteration of c.c.c. posets to force that, for many ideals in that class, their associated cardinal invariants (i.e., additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line.