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

TL;DR

This paper demonstrates that the set of probability measures under which a certain martingale has the Martingale Representation Property is either empty or dense, with implications for financial economics and endogenous market completeness.

Contribution

It establishes a density result for measures with the MRP using analytic fields and measure-theoretic arguments, advancing understanding of market completeness.

Findings

The set of measures with MRP is either empty or dense in the L-infinity norm.

The property depends on the Lebesgue measure of a related set of points.

The results are motivated by endogenous completeness in financial markets.

Abstract

Let $\psi$ be a multi-dimensional random variable. We show that the set of probability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale $S^{\mathbb{Q}}_t=\mathbb{E}^{\mathbb{Q}}\left[\psi\lvert\mathcal{F}_{t}\right]$ 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 $S_t(x) = \mathbb{E}^{\mathbb{Q}(x)}\left[\psi(x)\lvert\mathcal{F}_{t}\right]$ 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.