On Jensen's inequality and H?lder's inequality for g-expectation

摘要

In this paper, we shall prove that for n > 1, the n-dimensional Jensen inequality holds for the g-expectation if and only if g is independent of y and linear with respect to z, in other words, the corresponding g-expectation must be linear. A Similar result also holds for the general nonlinear expectation defined in Coquet et al. (Prob. Theory Relat. Fields 123 (2002), 1-27 or Peng (Stochastic Methods in Finance Lectures, LNM 1856, 143-217, Springer-Verlag, Berlin, 2004). As an application of a special n-dimensional Jensen inequality for g-expectation, we give a sufficient condition for g under which the H?lder's inequality and Minkowski's inequality for the corresponding g-expectation hold true.