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

Journal article

Let ψ be a multidimensional random variable. We show that the set of probability measures Q such that the Q-martingale SQt=EQ[ψ|Ft] 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.

Authors: Kramkov, D; Pulido, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematical Statistics
• Journal: Annals of Probability
• Volume: 47
• Issue: 4
• Pages: 2563-2581
• Publication date: 2019-07-04
• Acceptance date: 2018-10-29
• DOI: 10.1214/18-AOP1321
• EISSN: 2168-894X
• ISSN: 0091-1798
• Keywords: martingales; martingale representation property; endogenous completeness; analytic fields; complete market; equilibrium; stochastic integrals