Distributional compatibility for change of measures

In this paper, we characterise compatibility of distributions and probability measures on a measurable space. For a set of indices \(\mathcal{J}\), we say that the tuples of probability measures \((Q_{i})_{i\in \mathcal{J}} \) and distributions \((F_{i})_{i\in \mathcal{J}} \) are compatible if there exists a random variable having distribution \(F_{i}\) under \(Q_{i}\) for each \(i\in \mathcal{J}\). We first establish an equivalent condition using conditional expectations for general (possibly uncountable) \(\mathcal{J}\). For a finite \(n\), it turns out that compatibility of \((Q_{1},\dots ,Q_{n})\) and \((F_{1},\dots ,F _{n})\) depends on the heterogeneity among \(Q_{1},\dots ,Q_{n}\) compared with that among \(F_{1},\dots ,F_{n}\). We show that under an assumption that the measurable space is rich enough, \((Q_{1},\dots ,Q_{n})\) and \((F_{1},\dots ,F_{n})\) are compatible if and only if \((Q_{1},\dots ,Q _{n})\) dominates \((F_{1},\dots ,F_{n})\) in a notion of heterogeneity order, defined via the multivariate convex order between the Radon–Nikodým derivatives of \((Q_{1},\dots ,Q_{n})\) and \((F_{1},\dots ,F_{n})\) with respect to some reference measures. We then proceed to generalise our results to stochastic processes, and conclude the paper with an application to portfolio selection problems under multiple constraints.