Jensen's inequality for backward SDEs driven by G-Brownian motion

摘要

In this note, we consider Jensen's inequality for the nonlinear expectation associated with backward SDEs driven by G-Brownian motion (G-BSDEs for short). At first, we give a necessary and sufficient condition for G-BSDEs under which the one-dimensional Jensen inequality holds. Second, we prove that for n > 1, the n-dimensional Jensen inequality holds for any nonlinear expectation if and only if the nonlinear expectation is linear, which is essentially due to Jia (Arch Math 94:489-499, 2010). As a consequence, we give a necessary and sufficient condition for G-BSDEs under which the n-dimensional Jensen inequality holds.