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 J $\mathcal{J}$ , we say that the tuples of probability measures ( Q i ) i ∈ J $(Q_{i})_{i\in \mathcal{J}} $ and distributions ( F i ) i ∈ J $(F_{i})_{i\in \mathcal{J}} $ are compatible if there exists a random variable having distribution F i $F_{i}$ under Q i $Q_{i}$ for each i ∈ J $i\in \mathcal{J}$ . We first establish an equivalent condition using conditional expectations for general (possibly uncountable) J $\mathcal{J}$ . For a finite n $n$ , it turns out that compatibility of ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q_{n})$ and ( F 1 , … , F n ) $(F_{1},\dots ,F _{n})$ depends on the heterogeneity among Q 1 , … , Q n $Q_{1},\dots ,Q_{n}$ compared with that among F 1 , … , F n $F_{1},\dots ,F_{n}$ . We show that under an assumption that the measurable space is rich enough, ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q_{n})$ and ( F 1 , … , F n ) $(F_{1},\dots ,F_{n})$ are compatible if and only if ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q _{n})$ dominates ( F 1 , … , F n ) $(F_{1},\dots ,F_{n})$ in a notion of heterogeneity order, defined via the multivariate convex order between the Radon–Nikodým derivatives of ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q_{n})$ and ( F 1 , … , F n ) $(F_{1},\dots ,F_{n})$ 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.