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.
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.
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@article{CRMATH_2003__337_11_725_0, author = {Chen, Zengjing and Kulperger, Reg and Jiang, Long}, title = {Jensen's inequality for \protect\emph{g}-expectation: part 1}, journal = {Comptes Rendus. Math\'ematique}, pages = {725--730}, publisher = {Elsevier}, volume = {337}, number = {11}, year = {2003}, doi = {10.1016/j.crma.2003.09.017}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2003.09.017/} }
TY - JOUR AU - Chen, Zengjing AU - Kulperger, Reg AU - Jiang, Long TI - Jensen's inequality for g-expectation: part 1 JO - Comptes Rendus. Mathématique PY - 2003 SP - 725 EP - 730 VL - 337 IS - 11 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2003.09.017/ DO - 10.1016/j.crma.2003.09.017 LA - en ID - CRMATH_2003__337_11_725_0 ER -
%0 Journal Article %A Chen, Zengjing %A Kulperger, Reg %A Jiang, Long %T Jensen's inequality for g-expectation: part 1 %J Comptes Rendus. Mathématique %D 2003 %P 725-730 %V 337 %N 11 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2003.09.017/ %R 10.1016/j.crma.2003.09.017 %G en %F CRMATH_2003__337_11_725_0
Chen, Zengjing; Kulperger, Reg; Jiang, Long. Jensen's inequality for g-expectation: part 1. Comptes Rendus. Mathématique, Tome 337 (2003) no. 11, pp. 725-730. doi : 10.1016/j.crma.2003.09.017. http://archive.numdam.org/articles/10.1016/j.crma.2003.09.017/
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