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Jensen’s inequality for backward stochastic differential equations. (English) Zbl 1109.60042
Summary: Under the Lipschitz assumption and square integrable assumption on $g$, the author proves that Jensen’s inequality holds for backward stochastic differential equations with generator $g$ if and only if $g$ is independent of $y,g(t,0)\equiv 0$ and $g$ is super-homogeneous with respect to $z$. This result generalizes the known results on Jensen’s inequality for $g$-expectation in [{\it P. Briand}, {\it F. Coquet}, {\it Y. Hu}, {\it J. Mémin} and {\it S. Peng}, Electron. Commun. Probab. 5, 101--117 (2000; Zbl 0966.60054); {\it Z. Chen}, {\it R. Kulperger} and {\it L. Jiang}, C. R. Math., Acad. Sci. Paris 337, No. 11, 725--730 (2003; Zbl 1031.60014) and ibid. 337, No. 12, 797--800 (2003; Zbl 1031.60015); {\it L. Jiang} and {\it Z. Chen}, Chin. Ann. Math., Ser. B 25, No. 3, 401--412 (2004; Zbl 1062.60057)].

MSC:
 60H10 Stochastic ordinary differential equations 60E15 Inequalities in probability theory; stochastic orderings
Full Text:
References:
 [1] Peng, S., Backward stochastic differential equations and related g-expectation, Backward Stochastic Dif- ferential Equations, N. El. Karoui and L. Mazliak (eds.), Pitman Research Notes in Math. Series, No. 364, Longman Harlow, 1997, 141--159. · Zbl 0892.60066 [2] Chen, Z. and Epstein, L., Ambiguity, risk and asset returns in continuous time, Econometrica, 70, 2002, 1403--1443. · Zbl 1121.91359 · doi:10.1111/1468-0262.00337 [3] Rosazza, G. E., Some examples of risk measure via g-expectations, Working Paper, Università di Milano Bicocca, Italy, 2004. [4] Briand, P., Coquet, F., Hu, Y., Mémin, J. and Peng, S., A converse comparison theorem for BSDEs and related properties of g-expectation, Electron. Comm. Probab., 5, 2000, 101--117. · Zbl 0966.60054 [5] Coquet, F., Hu, Y., Mémin, J. and Peng, S., A general converse comparison theorem for backward stochas- tic differential equations, C. R. Acad. Sci. Paris, Série I, 333(7), 2001, 577--581. · Zbl 0994.60064 [6] Coquet, F., Hu, Y., Mémin, J. and Peng, S., Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory and Related Fields, 123(1), 2002, 1--27. · Zbl 1007.60057 · doi:10.1007/s004400100172 [7] Chen, Z., Kulperger, R. and Jiang, L., Jensen’s inequality for g-expectation: Part 1, C. R. Acad. Sci. Paris, Série I, 337(11), 2003, 725--730. · Zbl 1031.60014 [8] Chen, Z., Kulperger, R. and Jiang, L., Jensen’s inequality for g-expectation: Part 2, C. R. Acad. Sci. Paris, Série I, 337(12), 2003, 797--800. · Zbl 1031.60015 [9] Jiang, L. and Chen, Z., On Jensen’s inequality for g-expectation, Chin. Ann. Math., 25B(3), 2004, 401--412. · Zbl 1062.60057 · doi:10.1142/S0252959904000378 [10] Jiang, L., A property of g-expectation, Acta Math. Sinica, English Series, 20(5), 2004, 769--778. · Zbl 1065.60065 · doi:10.1007/s10114-004-0377-4 [11] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Let., 14, 1990, 55--61. · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6 [12] Jiang, L., Nonlinear Expectation--g-Expectation Theory and Its Applications in Finance, Doctoral Dis- sertation, Shandong University, China, 2005.