Abstract :
In this paper, the general filtration consistent nonlinear expectation defined on the
integrable variable space is considered, based on the results in [F. Coquet, Y. Hu, J. Memin,
S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab.
Theory Related Fields 123 (2002) 1–27]. Under a natural continuous assumption for the
nonlinear expectation, which weakens the domination assumption in [F. Coquet, Y. Hu,
J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation,
Probab. Theory Related Fields 123 (2002) 1–27], the author obtains the necessary and
sufficient conditions under which Jensen’s inequality for filtration consistent nonlinear
expectation holds in general, respectively on scalar function and bivariate function. These
two results generalize the known results on Jensen’s inequality for g-expectation in
[Z. Chen, R. Kulperger, L. Jiang, Jensen’s inequality for g-expectation: Part 1, C. R. Acad.
Sci. Paris Ser. I 337 (11) (2003) 725–730; Z. Chen, R. Kulperger, L. Jiang, Jensen’s inequality
for g-expectation: Part 2, C. R. Acad. Sci. Paris Ser. I 337 (12) (2003) 797–800; L. Jiang, On
Jensen’s inequality of bivariate function for g-expectation, J. Shandong Univ. 38 (5) (2003)
13–22 (in Chinese); L. Jiang, Z. Chen, On Jensen’s inequality for g-expectation, Chinese Ann.
Math. Ser. B 25 (3) (2004) 401–412; L. Jiang, Jensen’s inequality for backward stochastic
differential equation, Chinese Ann. Math. Ser. B 27 (5) (2006) 553–564; S. Fan, Jensen’s
inequality for g-expectation on convex (concave) function, Chinese Ann. Math. Ser. A 27
(5) (2006) 635–644 (in Chinese)].
