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山东大学学报(理学版) ›› 2016, Vol. 51 ›› Issue (8): 1-9.doi: 10.6040/j.issn.1671-9352.0.2016.312

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基于加权g-期望的Jensen不等式,矩不等式与大数定律

江龙,陈敏   

  1. 中国矿业大学理学院, 江苏 徐州 221116
  • 收稿日期:2016-06-29 出版日期:2016-08-20 发布日期:2016-08-08
  • 作者简介:江龙(1964— ), 男, 博士, 教授, 研究方向为倒向随机微分方程与非线性数学期望. E-mail:jianglong365@hotmail.com
  • 基金资助:
    国家自然科学基金资助项目(11371362)

Jensens inequality, moment inequality and law of large numbers for weighted g-expectation

JIANG Long, CHEN Min   

  1. College of Sciences, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China
  • Received:2016-06-29 Online:2016-08-20 Published:2016-08-08

摘要: 在g-期望的基础上提出加权g-期望ελg [·]的概念。证明了当生成元g关于y非增且关于(y,z)满足正齐次性时, 基于加权 g-期望的矩不等式一般成立。 在λ≥1/2 且生成元g不依赖于y的条件下, 在g关于z满足超齐次性时, 建立了基于加权g-期望的Jensen不等式; 当g关于z满足次线性时, 建立了基于加权g-期望的大数定律。

关键词: g-期望, Jensen不等式, 大数定律, 矩不等式, 加权g-期望

Abstract: We propose the notion of weighted g-expectation ελg [·] based on g-expectation. We prove that if the generator g is non-increasing with respect to y and positive-homogeneous with respect to (y,z), the moment inequality for weighted g-expectation holds in general. When λ≥1/2 and the generator g is independent of y, we establish Jensens inequality for weighted g-expectation when g is super-homogeneous with respect to z, and we establish the law of large numbers for weighted g-expectation when g is sublinear with respect to z.

Key words: weighted g-expectation, Jensens inequality, law of large numbers, g-expectation, moment inequality

中图分类号: 

  • O211
[1] PARDOUX E, PENG S G. Adapted solution of a backward stochastic differential equation[J]. Systems Control Letters, 1990, 14:55-61.
[2] PENG S G. Backward stochastic differential equations and related g-expectations[M] // Backward Stochastic Differential Equations, Karoui N. E. and Mazliak L., eds., Pitman Research Notes in Math., Series 364, London: Longman Harlow, 1997: 141-159.
[3] BRIAND P, COQUET F, HU Y, et al. A converse comparison theorem for BSDEs and related properties of g-expectation[J]. Elect Comm Probab, 2000, 5:101-117.
[4] JIANG L. Convexity, translation invariance and subadditivity for g-expectations and related risk measures[J]. The Annals of Applied Probability, 2008: 245-258.
[5] JIANG L. Jensens Inequality for backward stochastic differential equations[J]. Chinese Annals of Mathematics, Series B, 2006, 27(5):553-564.
[6] COQUET F, HU Y, MÉMIN J, et al. Filtration-consistent nonlinear expectations and related g-expectations[J]. Probability Theory and Related Fields, 2002, 123(1):1-27.
[7] JIA G, PENG S. Jensen.s inequality for g-convex function under g-expectation[J]. Probability Theory and Related Fields, 2010, 147(1-2):217-239.
[8] JI S, ZHOU X Y. A generalized neyman pearson iemma for g-probabilities[J]. Probability Theory and Related Fields, 2010, 148(3-4):645-669.
[9] 杨丛,江龙.关于g-期望的几个不等式[J].华东师范大学学报:自然科学版,2013(2):111-115. YANG Cong, JIANG Long. Several inequalities of the g-expectation [J]. Journal of East China Normal University(Natural Science), 2013(2):111-115.
[10] LIN Q, SHI Y F. Law of large numbers for Peng g-expectation[J]. Scientia Sinica(Mathematica), 2012, 42(4):295-302.
[11] JIANG L. A necessary and suffcient condition for probability measures dominated by g-expectation[J]. Statistics and Probability Letters, 2009, 79(2):196-201.
[12] CHEN Z, KULPERGER R, JIANG L. Jensens inequality for g-expectation: part 1[J]. Comptes Rendus Mathematique, 2003, 337(11):725-730.
[13] JIANG L, CHEN Z. On Jensens inequality for g-expectation[J]. Chinese Annals of Mathematics, 2004, 25(03):401-412.
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