Conditional nonlinear expectations

Abstract

Let Ω be a Polish space with Borel σ-field F and countably generated sub σ-field G⊂F. Denote by L(F) the set of all bounded F-upper semianalytic functions from Ω to the reals and by L(G) the subset of G-upper semianalytic functions. Let E(⋅|G):L(F)→L(G) be a sublinear increasing functional which leaves L(G) invariant. It is shown that there exists a G-analytic set-valued mapping PG from Ω to the set of probabilities which are concentrated on atoms of G with compact convex values such that E(X|G)(ω)=supP∈PG(ω)EP[X] if and only if E(⋅|G) is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional E(⋅):L(F)→R which leaves the constants invariant, the tower property E(⋅)=E(E(⋅|G)) is characterized via a pasting property of the representing sets of probabilities, and the importance of analytic functions is explained. Finally, it is characterized when a nonlinear version of Fubini’s theorem holds true and when the product of a set of probabilities and a set of kernels is compact.