Jensen's inequality for filtration consistent nonlinear expectation without domination condition
A nonlinear expectation is a functional \(\mathcal{E}\) defined on real integrable variables \(\xi\), which preserves the order, which preserves the constants, but which is generally not linear. If moreover one is given a filtration \(\mathcal{F}_t\), the nonlinear expectation is said to be filtration consistent if conditional expectations \(\mathcal{E}[\xi|\mathcal{F}_t]\) exist. The aim of this article is to study Jensen's inequality for these conditional expectations. This problem has already been studied for \(g\)-expectations, which are a particular class of nonlinear expectations (this particular class can be studied with the help of backward stochastic differential equations); here the general case is considered. The main result states that Jensen's inequality on \(\mathcal{E}[\phi(\xi)|\mathcal{F}_t]\) holds for any convex function \(\phi\) (such that \(\xi\) and \(\phi(\xi)\) are integrable) if and only if it holds for affine functions. Necessary and sufficient conditions for Jensen's inequality on \(\mathcal{E}[\phi(\xi,\eta)|\mathcal{F}_t]\) are also given.
filtration consistent nonlinear expectation
scientific article; zbMATH DE number 5299471