Density of the set of probability measures with the martingale representation property

Abstract

Let $\psi$ be a multidimensional random variable. We show that the set of probability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale $S^{\mathbb{Q}}_{t}=\mathbb{E}^{\mathbb{Q}}[\psi \lvert\mathcal{F}_{t}]$ has the Martingale Representation Property (MRP) is either empty or dense in $\mathcal{L}_{\infty}$-norm. The proof is based on a related result involving analytic fields of terminal conditions $(\psi(x))_{x\in U}$ and probability measures $(\mathbb{Q}(x))_{x\in U}$ over an open set $U$. Namely, we show that the set of points $x\in U$ such that $S_{t}(x)=\mathbb{E}^{\mathbb{Q}(x)}[\psi(x)\lvert\mathcal{F}_{t}]$ does not have the MRP, either coincides with $U$ or has Lebesgue measure zero. Our study is motivated by the problem of endogenous completeness in financial economics.