Abstract
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 Hölder’s inequality and Minkowski’s inequality for the corresponding g-expectation hold true.
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The author thanks the partial support from the National Basic Research Program of China (973 Program) grant No. 2007CB814901 (Financial Risk) and the National Natural Science Foundation of China, grant No. 10671111.
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Jia, G. On Jensen’s inequality and Hölder’s inequality for g-expectation. Arch. Math. 94, 489–499 (2010). https://doi.org/10.1007/s00013-010-0117-1
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DOI: https://doi.org/10.1007/s00013-010-0117-1