DENSITY OF THE SET OF PROBABILITY MEASURES WITH THE MARTINGALE REPRESENTATION PROPERTY

Let ψ be a multidimensional random variable. We show that the set of probability measures ℚ such that the ℚ-martingale S t ℚ = E ℚ [ ψ | F t ] has the Martingale Representation Property (MRP) is either empty or dense in L∞ -norm. The proof is based on a related result involving analytic fields of terminal conditions (ψ(x))x∈U and probability measures (ℚ(x))x∈U over an open set U. Namely, we show that the set of points x ∈ U such that St(x) = Eℚ(x)[ψ(x)|Ft] 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.