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.
References
Alvarez O., Lasry J.-M., Lions P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. 76, 265–288 (1997)
Briand P., Coquet F., Hu Y., Mémin J., Peng S.: A converse comparison theorem for BSDEs and related properties of g-expectation. Electron. Commun. Probab. 5, 101–117 (2000)
Barrieu, P., El Karoui, N.: Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Carmona, R. (ed.) Volume on Indifference Pricing (preprint). Princeton University Press, New Jersey (2005, to appear)
Buckdahn R., Quincampoix M., Răşcanu A.: Viability property for a backward stochastic differential equations and applications to partial differential equations. Probab. Theory Relat. Fields 116, 485–504 (2000)
Chen Z., Epstein L.: Ambiguity, risk andasset returns in continuous time. Econometrica 70, 1403–1443 (2002)
Chen Z., Kulperger R., Jiang L.: Jensen’s inequality for g-expectation: Part 1. C. R. Acad. Sci. Paris Ser. I 333, 725–730 (2003)
Coquet F., Hu Y., Mémin J., Peng S.: Filtration consistent nonlinear expectations and related g-expectations. Probab. Theory Relat. Fields 123, 1–27 (2002)
Crandall M. G., Ishii H., Lions P.L.: Users’ guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)
Duffie D., Epstein L.: Stochastic differential utility. Econometrica 60(2), 353–394 (1992)
El Karoui N., Peng S., Quenez M.C.: Backward stochastic differential equations in finance. Math. Finance 7(1), 1–71 (1997)
El Karoui, N., Quenez, M.C.: Nonlinear pricing theory and backward stochastic differential equations. Biais, B., et al. (eds.) Financial mathematics. Letures given at the 3rd Session of the Centro Internazionale Matematico Estivo (CIME), held in Bressanone, Italy, 8–13 July 1996. Lect. Notes Math., vol. 1656, pp. 191–246. Springer, Berlin (1997)
Frittelli, M., Rossaza Gianin, E.: Dynamic convex risk measures, Szegö (ed.) Risk Measures for the 21st Century, pp. 227–247. Wiley-Finance (2004)
Hiriart-Urruty J.-B., Lemaréchal C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1991)
Hu Y.: On Jensen’s inequality for g-expectation and for nonlinear expectation. Archiv der Mathematik 85, 572–580 (2005)
Jiang L.: Jensen’s inequality for backward stochastic differential equations. Chin. Ann. Math. 27B(5), 553–564 (2006)
Pallaschke D., Rolewicz S.: Foundations of Mathematical Optimization. Kluwer, Dordrecht (1997)
Pardoux E., Peng S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)
Peng, S.: BSDE and stochastic optimizations, topics in stochastic analysis. In: Yan, J., Peng, S., Fang, S., Wu, L.M. (eds.) Lecture Notes of 1995 Summer School in Math, Chap. 2, (Chinese vers.). Science Press, Beijing (1997)
Peng, S.: BSDE and related g-expectation. In: El Karoui, N., Mazliak, L. (eds.) Pitman Research Notes in Mathematics Series, no. 364. Backward Stochastic Differential Equations, pp. 141–159 (1997)
Peng S.: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab. Theory Relat. Fields 113(4), 473–499 (1999)
Peng, S.: Nonlinear expectation, nonlinear evaluations and risk measurs. In: Back, K., Bielecki, T.R., Hipp, C., Peng, S., Schachermayer, W. (eds.) Stochastic Methods in Finance Lectures. LNM, vol. 1856, pp. 143–217. Springer, Heidelberg (2004)
Peng S.: Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. Engl. Ser. 20(2), 1–24 (2004)
Peng S.: Nonlinear expectations and nonlinear Markov chains. Chin. Ann. Math. 26(2), 159–184 (2005)
Peng S.: Dynamical evaluations. C. R. Acad. Sci. Paris Ser. I 339, 585–589 (2005)
Peng, S.: Dynamically consistent nonlinear evaluations and expectations, preprint (pdf-file available in arXiv:math.PR/0501415 v1 24 Jan 2005) (2005)
Peng, S.: G-Expectation, G-Brownian motion and related stochastic calculus of Itô’s type. In: Benth, F.E., et al. (eds.) Stochastic Analysis and Applications, The Abel Symposium 2005, Proceedings of the second Abel Symposium, pp 541–567. Springer-Verlag (2006)
Peng, S.: Modelling Derivatives Pricing Mechanisms with Their Generating Functions, preprint (pdf-file available in arXiv:math.PR/0605599v1 23 May 2006) (2006)
Rosazza-Gianin E.G.: Risk measures via g-expectations. Insur. Math. Econ. 36(1), 19–34 (2004)
Singer I.: Abstract Convex Analysis. Wiley, New York (1997)
Yong J.: European-type contingent claims in an incomplete market with constrained wealth and portfolio. Math. Finance 9(4), 387–412 (1999)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0206-x
Keywords
- Backward stochastic differential equation
- Backward stochastic viability property
- g-Convexity
- g-Expectation
- Jensen’s inequality
- Viscosity subsolution
Mathematics Subject Classification (2000)
- 60H10