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

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 te...

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Veröffentlicht in:IDEAS Working Paper Series from RePEc 2019-07, Vol.47 (4), p.2563-2581
Hauptverfasser: Kramkov, Dmitry, Pulido, Sergio
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General Finance
Mathematics
Probability
Quantitative Finance
Random variables
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description 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.
doi_str_mv 10.1214/18-AOP1321
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subjects General Finance
Mathematics
Probability
Quantitative Finance
Random variables
title DENSITY OF THE SET OF PROBABILITY MEASURES WITH THE MARTINGALE REPRESENTATION PROPERTY
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