Comptes Rendus
no. 9-10
Probability Theory
On the integral representation of g-expectations
[Sur le représentation intégrale pour les g-espérances]

Présenté par : Marc Yor

Mingshang Hu1
1 School of Mathematics, Shandong University, Jinan 250100, China
Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 571-574.

Dans cette Note, nous donnons une condition nécessaire et suffisante sur g déterministe sous laquelle les g-espérances peut être représentée par les espérances de Choquet.

In this Note, we give a necessary and sufficient condition on deterministic g under which g-expectations can be represented as Choquet expectations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.04.008
Mingshang Hu 1

1 School of Mathematics, Shandong University, Jinan 250100, China 
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Mingshang Hu. On the integral representation of g-expectations. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 571-574. doi : 10.1016/j.crma.2010.04.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.008/

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

[2] Z. Chen; T. Chen; M. Davison Choquet expectation and Peng's g-expectation, The Annals of Probability, Volume 33 (2005) no. 3, pp. 1179-1199

[3] Z. Chen; L. Epstein Ambiguity, risk and asset returns in continuous time, Econometrica, Volume 70 (2002), pp. 1403-1443

[4] Z. Chen; R. Kulperger Minimax pricing and Choquet pricing, Insurance: Mathematics and Economics, Volume 38 (2006), pp. 518-528

[5] Z. Chen; A. Sulem An integral representation theorem of g-expectations, Research Report INRIA, Volume 4284 (2001), pp. 1-20

[6] G. Choquet Theory of capacities, Ann. Inst. Fourier (Grenoble), Volume 5 (1953), pp. 131-195

[7] F. Coquet; Y. Hu; J. Mémin; S. Peng Filtration consistent nonlinear expectations and related g-expectations, Probability Theory and Related Fields, Volume 123 (2002), pp. 1-27

[8] D. Denneberg Non-additive Measure and Integral, Kluwer Academic Publishers, Boston, 1994

[9] N. El Karoui; S. Peng; M.C. Quenez Backward stochastic differential equations in finance, Mathematical Finance, Volume 7 (1997), pp. 1-71

[10] M. Hu Choquet expectations and g-expectations with multi-dimensional Brownian motion, 2009 (available via) | arXiv

[11] L. Jiang Convexity, translation invariance and subadditivity for g-expectations and related risk measures, Annals of Applied Probability, Volume 18 (2008) no. 1, pp. 245-258

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

[13] S. Peng BSDE and stochastic optimizations, topics in stochastic analysis (J. Yan; S. Peng; S. Fang; L.M. Wu, eds.), Lecture Notes of 1995 Summer School in Mathematics, Science Press, Beijing, 1997 Ch. 2 (Chinese vers.)

[14] S. Peng Backward SDE related g-expectations, Backward stochastic differential equations (N. El Karoui; L. Mazliak, eds.), Pitman Research Notes in Mathematics Series, vol. 364, Longman, Harlow, 1997, pp. 141-159

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