# Distributional Compatibility for Change of Measures

## TL;DR

This paper characterizes when multiple probability measures and distributions are compatible by using heterogeneity orders and extends the results to stochastic processes, with applications to portfolio selection under constraints.

## Contribution

It introduces a new heterogeneity order criterion for compatibility of measures and distributions, generalizes to stochastic processes, and applies to portfolio optimization problems.

## Key findings

- Compatibility depends on heterogeneity comparison of measures and distributions.
- Compatibility characterized by dominance in heterogeneity order via convex order.
- Results extend to stochastic processes with practical portfolio applications.

## Abstract

In this paper, we characterize 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 multivariate convex order between the Radon-Nikodym derivatives of $(Q_1,\dots,Q_n)$ and $(F_1,\dots,F_n)$ with respect to some reference measures. We then proceed to generalize our results to stochastic processes, and conclude the paper with an application to portfolio selection problems under multiple constraints.