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

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.
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Submitted on : Thursday, August 8, 2019 - 5:27:23 PM
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Dmitry Kramkov, Sergio Pulido. Density of the set of probability measures with the martingale representation property. Annals of Probability, Institute of Mathematical Statistics, 2019, 47 (4), pp.2563-2581. ⟨10.1214/18-AOP1321⟩. ⟨hal-01598651v2⟩

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