## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2563-2581.

Dates
Revised: July 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205716

Digital Object Identifier
doi:10.1214/18-AOP1321

Mathematical Reviews number (MathSciNet)
MR3980928

#### Citation

Kramkov, Dmitry; Pulido, Sergio. Density of the set of probability measures with the martingale representation property. Ann. Probab. 47 (2019), no. 4, 2563--2581. doi:10.1214/18-AOP1321. https://projecteuclid.org/euclid.aop/1562205716

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