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

Dmitry Kramkov () and Sergio Pulido ()
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Dmitry Kramkov: CMU - Carnegie Mellon University [Pittsburgh]
Sergio Pulido: ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise, LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - INRA - Institut National de la Recherche Agronomique - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - CNRS - Centre National de la Recherche Scientifique

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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.

Keywords: martingale representation property; martingales; stochastic integrals; analytic fields; endogenous completeness; complete market; equilibrium (search for similar items in EconPapers)
Date: 2019-07
Note: View the original document on HAL open archive server: https://hal.science/hal-01598651v2
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Citations: View citations in EconPapers (1)

Published in Annals of Probability, 2019, 47 (4), pp.2563-2581. ⟨10.1214/18-AOP1321⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01598651

DOI: 10.1214/18-AOP1321

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