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July 2019 Density of the set of probability measures with the martingale representation property
Dmitry Kramkov, Sergio Pulido
Ann. Probab. 47(4): 2563-2581 (July 2019). DOI: 10.1214/18-AOP1321

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.

Citation

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Dmitry Kramkov. Sergio Pulido. "Density of the set of probability measures with the martingale representation property." Ann. Probab. 47 (4) 2563 - 2581, July 2019. https://doi.org/10.1214/18-AOP1321

Information

Received: 1 September 2017; Revised: 1 July 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114724
MathSciNet: MR3980928
Digital Object Identifier: 10.1214/18-AOP1321

Subjects:
Primary: 60G44, 60H05, 91B51, 91G99

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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