Stochastic Partial Ordering

Abstract

A probability measure $P$ on a partially ordered Polish space $E$ is called stochastically smaller than $Q$ (notation: $P \leqslant Q$) if $\int f dP \leqslant \int f dQ$ holds for all bounded increasing measurable $f$. We investigate the question when for a stochastically increasing family $\{P_t, t \in \mathbb{R}\}$ there exists an increasing process $\{X_t, t \in \mathbb{R}\}$ with 1-dimensional marginal distributions $P_t$. A sufficient condition, satisfied, e.g., for $E = \mathbb{R}^\mathbf{N}$, for compact $E$ and for spaces $E$ of Lipschitz-functions, is the compactness of all intervals $\{z \in E: x \leqslant z \leqslant y\}$; but for general countable $E$ such an increasing $E$-valued process $\{X_t\}$ need not exist.