ON JENSEN’S INEQUALITY FOR g-EXPECTATION

Briand et al. gave a counterexample showing that given g, Jensen's inequalityfor g-expectation usually does not hold in general. This paper proves that Jensen'sinequality for g-expectation holds in general if and only if the generator g(t,z) issuper-homogeneous in z. In particular, g is not necessarily convex in z.     

g-期望关于凸(凹)函数的Jensen不等式

在文[8]的基础上和彭实戈提出的关于g-期望的最基本的条件下,证明了g-期望关于凸(凹)函数的Jensen不等式在一般意义下成立当且仅当g是关于(y,z)的超齐次(次齐次)生成元且不依赖于y．     

关于几何凸函数的Jensen不等式的加细

利用几何凸函数的Jensen不等式建立一个由{1,2,…,n}到(0,+∞)上的一个映射,研究了这个映射的单调性,获得一个该Jensen不等式的加细,并得到几何凸函数的一些新的不等式.     

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)].     

涉及Jensen函数的不等式

引入了Jensen函数及Jensen平均的概念,借助于数学分析和代数工具给出了Jensen函数的分解公式,利用这个公式给出了推广和加强Jensen不等式的一种崭新的思路,作为应用,给出了Jensen不等式成立的一个有趣的充分条件．旨在为数学研究提供一些有用的解析不等式．     

Superquadratic functions in several variables

The concept of superquadratic functions in several variables, as a generalization of the same concept in one variable is introduced. Analogous results to results obtained for convex functions in one and several variables are presented. These include refinements of Jensen's inequality and its counterpart, and of Slater-Pe?ari?'s inequality.     

Jensen's inequality relative to matrix-valued measures

A Bochner-integral formulation of Jensen's inequality is presented for Hermitian matrix-valued functions and measures.     

On a global upper bound for Jensen's inequality

In this paper we shall give a global upper bound for Jensen's inequality without restrictions on the target convex function f. We also introduce a characteristic c(f) i.e. an absolute constant depending only on f, by which the global bound is improved.     

A note on Jensen’s inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.     

非线性Sobolev-Galpern方程解的blow up

本文研究非线性Ｓｏｂｏｌｅｖ－Ｃａｌｐｅｒｎ方程的初边值问题整体解的不存性即解的爆破问题,用能量估计方法并借助于Ｊｅｎｓｅｎ不等式证明了非线性Ｓｏｂｏｌｉｖ－Ｇａｌｐｅｒｎ方程各种初边值问题在某些假设下不存在整体解。     

Jensen’s Inequality for Backward Stochastic Differential Equations

Abstract Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen’s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) ≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen’s inequality for g- expectation in [4, 7–9]. *Project supported by the National Natural Science Foundation of China (No.10325101) and the Science Foundation of China University of Mining and Technology.     

Moment inequality and Hölder inequality for BSDEs

Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.     

ON JENSEN’S INEQUALITY FOR g-EXPECTATION

Briand et al.gave a counterexample showing that given g, Jensen‘s inequality for g-expectation usually does not hold in general. This paper proves that Jensen‘s inequality for g-expectation holds in general if and only if the generator g(t,z) is super-homogeneous in z. In particular, g is not necessarily convex in z.     

Sufficiency and Jensen's inequality for conditional expectations

For finite sets of probability measures, sufficiency is characterized by means of certain positively homogeneous convex functions. The essential tool is a discussion of equality in Jensen's inequality for conditional expectations. In particular, it is shown that characterizations of sufficiency by Csiszár's f-divergence (1963, Publ. Math. Inst. Hung. Acad. Sci. Ser. A, 8, 85–107) and by optimal solutions of a Bayesian decision problem used by Morse and Sacksteder (1966, Ann. Math. Statist., 37, 203–214) can be proved by the same method.     

类似Hilbert不等式的一些新不等式的推广

本文推广了 Pachpatte在文 [1 ]中给出的类似 Hilbert不等式的一些新不等式 .     

Jensen's Operator Inequality

Jensen's operator inequality and Jensen's trace inequality forreal functions defined on an interval are established in whatmight be called their definitive versions. This is accomplishedby the introduction of genuine non-commutative convex combinationsof operators, as opposed to the contractions considered in earlierversions of the theory by the authors, and by Brown and Kosaki.As a consequence, one no longer needs to impose conditions onthe interval of definition. It is shown how this relates tothe pinching inequality of Davis, and how Jensen's trace inequalitygeneralizes to C*-algebras. 2000 Mathematics Subject Classification47A56 (primary), 46L10, 47C15, 26A51 (secondary).     

Operator Jensen＇s Inequality on C＊-algebras

A real continuous function which is defined on an interval is said to beA-convex if it is convex on the set of self-adjoint elements,with spectra in the interval,in all matrix algebras of the unital C-algebra A.We give a general formation of Jensen’s inequality for A-convex functions.     

离散和连续型Pachpatte不等式的逆

本文利用分析的方法和不等式理论,建立了离散型和连续型Pachpatte不等 式的逆向不等式.作为应用,推广并改进了一些新型Hilbert不等式.     

Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Let (T, , P) be a probability space, a P-complete sub-δ-algebra of and X a Banach space. Let multifunction t → Γ(t), t T, have a (X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε E_p x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = , then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.     

关于Banach空间中的向量值的Bochner积分型的广义Jensen不等式

李泽民(1990)将R^n中的极值问题的Kuhu-Tucker条件推广到了线性拓扑空间中的向量极值问题．本文作者从另一角度，以锥为工具，把在概率论与鞅论等学科有着广泛应用的R中的著名的Jensen不等式推广到序Banach空间，导出向量值的Bochner积分型的广义Jensen不等式，从而推广了前人的工作．