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

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.Comment: 24 pages, forthcoming in Annals of Probabilit

On the Second Fundamental Theorem of Asset Pricing

Let $X^1,\ldots, X^d$ be sigma-martingales on $(\Omega,{\cal F}, P)$. We show that every bounded martingale (with respect to the underlying filtration) admits an integral representation w.r.t. $X^1,\ldots, X^d$ if and only if there is no equivalent probability measure (other than $P$) under which $X^1,\ldots,X^d$ are sigma-martingales. From this we deduce the second fundamental theorem of asset pricing- that completeness of a market is equivalent to uniqueness of Equivalent Sigma-Martingale Measure (ESMM)

A theory of stochastic integration for bond markets

We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of super-replication and utility maximization from terminal wealth in a bond market. Finally, we compare our approach to those already existing in literature.Comment: Published at http://dx.doi.org/10.1214/105051605000000548 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org