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Jensen’s inequality for g-convex function under g-expectation
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  • Published: 27 February 2009

Jensen’s inequality for g-convex function under g-expectation

  • Guangyan Jia1 &
  • Shige Peng1 

Probability Theory and Related Fields volume 147, pages 217–239 (2010)Cite this article

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  • 13 Citations

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Abstract

A real valued function h defined on \({\mathbb{R}}\) is called g-convex if it satisfies the “generalized Jensen’s inequality” for a given g-expectation, i.e., \({h(\mathbb{E}^{g}[X])\leq \mathbb{E}^{g}[h(X)]}\) holds for all random variables X such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient condition for a C 2-function being g-convex, and study some more general situations. We also study g-concave and g-affine functions, and a relation between g-convexity and backward stochastic viability property.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, 250100, Jinan, Shandong, People’s Republic of China

    Guangyan Jia & Shige Peng

Authors
  1. Guangyan Jia
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  2. Shige Peng
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Corresponding author

Correspondence to Guangyan Jia.

Additional information

The authors thank the partial support from the National Basic Research Program of China (973 Program) grant No. 2007CB814900 and No. 2007CB814901 (Financial Risk). The first author also thanks the partial support from the National Natural Science Foundation of China, grant No. 10671111.

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Jia, G., Peng, S. Jensen’s inequality for g-convex function under g-expectation. Probab. Theory Relat. Fields 147, 217–239 (2010). https://doi.org/10.1007/s00440-009-0206-x

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  • Received: 19 February 2008

  • Revised: 30 January 2009

  • Published: 27 February 2009

  • Issue Date: May 2010

  • DOI: https://doi.org/10.1007/s00440-009-0206-x

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Keywords

  • Backward stochastic differential equation
  • Backward stochastic viability property
  • g-Convexity
  • g-Expectation
  • Jensen’s inequality
  • Viscosity subsolution

Mathematics Subject Classification (2000)

  • 60H10
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