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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)0 and g is super-homogeneous with respect to z. This result generalizes the known results on Jensen’s inequality for g-expectation in [P. Briand, F. Coquet, Y. Hu, J. Mémin and S. Peng, Electron. Commun. Probab. 5, 101–117 (2000; Zbl 0966.60054); Z. Chen, R. Kulperger and 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); L. Jiang and Z. Chen, Chin. Ann. Math., Ser. B 25, No. 3, 401–412 (2004; Zbl 1062.60057)].
MSC:
60H10Stochastic ordinary differential equations
60E15Inequalities in probability theory; stochastic orderings
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.
[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.
[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.
[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.
[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.
[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.