Conditional nonlinear expectations

Abstract

Let $\Omega$ be a Polish space with Borel $\sigma$-field $\mathcal{F}$ and countably generated sub $\sigma$-field $\mathcal{G}\subset\mathcal{F}$. Denote by $\mathcal{L}(\mathcal{F})$ the set of all bounded $\mathcal{F}$-upper semianalytic functions from $\Omega$ to the reals and by $\mathcal{L}(\mathcal{G})$ the subset of $\mathcal{G}$-upper semianalytic functions. Let $\mathcal{E}(\cdot|\mathcal{G})\colon\mathcal{L}(\mathcal{F})\to\mathcal{L}(\mathcal{G})$ be a sublinear increasing functional which leaves $\mathcal{L}(\mathcal{G})$ invariant. It is shown that there exists a $\mathcal{G}$-analytic set-valued mapping $\mathcal{P}_{\mathcal{G}}$ from $\Omega$ to the set of probabilities which are concentrated on atoms of $\mathcal{G}$ with compact convex values such that $\mathcal{E}(X|\mathcal{G})(\omega)=$ $\sup_{P\in\mathcal{P}_{\mathcal{G}}(\omega)} E_P[X]$ if and only if $\mathcal{E}(\cdot |\mathcal{G})$ is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional $\mathcal{E}(\cdot)\colon\mathcal{L}(\mathcal{F})\to\mathbb{R}$ which leaves the constants invariant, the tower property $\mathcal{E}(\cdot)=\mathcal{E}(\mathcal{E}(\cdot|\mathcal{G}))$ 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.