Jensen's inequality for <i>g</i>-expectation: part 1

Zengjing Chen; Reg Kulperger; Long Jiang

doi:10.1016/j.crma.2003.09.017

Probability Theory/Statistics

Jensen's inequality for g-expectation: part 1

Presented by: Paul Deheuvels

Zengjing Chen ¹; Reg Kulperger ²; Long Jiang ¹

¹ Department of Mathematics, Shandong University, Jinan, 250100, China
² Department of Statistical and Actuarial Science, The University of Western Ontario, London, Ontario, Canada

Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 725-730.

Abstract
Résumé

Briand et al. (Electron. Comm. Probab. 5 (2000) 101–117) gave a counterexample and proposition to show that given g,g-expectations usually do not satisfy Jensen's inequality for most of convex functions. This yields a natural question, under which conditions on g, do g-expectations satisfy Jensen's inequality for convex functions? In this paper, we shall deal with this question in the case that g is convex and give a necessary and sufficient condition on g under which Jensen's inequality holds for convex functions.

Briand et al. (Electron. Comm. Probab. 5 (2000) 101–117) ont donné un contre-exemple et une proposition qui démontrent que donné g, les g-espérances ne satisfont pas l'inégalité de Jensen pour la majorité des fonctions convexes. Ceci mène donc de façon naturelle à la question : sous quelles conditions sur g les g-espérances satisfont l'inégalité de Jensen pour les fonctions convexes ? Dans cet article, nous obtenons une solution pour un g convexe et donnons une condition nécessaire et suffisante sur g sous laquelle l'inégalité de Jensen est satisfaite pour tout les fonctions convexes.

Received: 2003-04-20
Accepted: 2003-09-17
Published online: 2003-11-07

DOI: 10.1016/j.crma.2003.09.017

Author's affiliations:

Zengjing Chen ¹; Reg Kulperger ²; Long Jiang ¹

¹ Department of Mathematics, Shandong University, Jinan, 250100, China
² Department of Statistical and Actuarial Science, The University of Western Ontario, London, Ontario, Canada

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     author = {Zengjing Chen and Reg Kulperger and Long Jiang},
     title = {Jensen's inequality for \protect\emph{g}-expectation: part 1},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {725--730},
     publisher = {Elsevier},
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     number = {11},
     year = {2003},
     doi = {10.1016/j.crma.2003.09.017},
     language = {en},
}

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Zengjing Chen; Reg Kulperger; Long Jiang. Jensen's inequality for g-expectation: part 1. Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 725-730. doi : 10.1016/j.crma.2003.09.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.017/

References
Cited by

[1] P. Briand; F. Coquet; Y. Hu; J. Mémin; S. Peng A converse comparison theorem for BSDEs and related properties of g-expectation, Electron. Comm. Probab., Volume 5 (2000), pp. 101-117

[2] Z. Chen; S. Peng A general downcrossing inequality for g-martingales, Statist. Probab. Lett., Volume 46 (2000), pp. 169-175

[3] E. Pardoux; S. Peng Adapted solution of a backward stochastic differential equation, Systems Control Lett., Volume 14 (1990), pp. 55-61

[4] S. Peng Backward stochastic differential equations and related g-expectation (N. El Karoui; L. Mazliak, eds.), Backward Stochastic Differential Equations, Pitman Res. Notes Math. Ser., 364, 1997, pp. 141-159

[5] S. Peng A generalized dynamic programming principle and Hamilton–Jacobi–Bellman equation, Stochastics, Volume 38 (1992) no. 2, pp. 119-134

[6] K. Yosida Functional Analysis, Springer-Verlag, Beijing, 1999

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