Density of the set of probability measures with the martingale representation property
Identifieur interne : 000004 ( France/Analysis ); précédent : 000003; suivant : 000005Density of the set of probability measures with the martingale representation property
Auteurs : Dmitry Kramkov [États-Unis] ; Sergio Pulido [France]Source :
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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.
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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.</div></front></TEI><affiliations><list><country><li>France</li><li>États-Unis</li></country></list><tree><country name="États-Unis"><noRegion><name sortKey="Kramkov, Dmitry" sort="Kramkov, Dmitry" uniqKey="Kramkov D" first="Dmitry" last="Kramkov">Dmitry Kramkov</name></noRegion></country><country name="France"><noRegion><name sortKey="Pulido, Sergio" sort="Pulido, Sergio" uniqKey="Pulido S" first="Sergio" last="Pulido">Sergio Pulido</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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