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 $S_t^{\mathbb{Q}}={\mathbb{E}}^{\mathbb{Q}}\left[\psi |{\mathcal{F}}_t\right]$ 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 $S_t(x)={\mathbb{E}}^{\mathbb{Q}(x)}\left[\psi (x)|{\mathcal{F}}_t\right]$ 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.

24 pages, forthcoming in Annals of Probability

References in corpus (5)

• On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets
• A system of quadratic BSDEs arising in a price impact model
• Existence of an endogenously complete equilibrium driven by a diffusion
• Integral representation of martingales motivated by the problem of endogenous completeness in financial economics
• Market Completion with Derivative Securities