The Annals of Probability

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

Dmitry Kramkov and Sergio Pulido

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Let $\psi$ be a multidimensional random variable. We show that the set of probability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale $S^{\mathbb{Q}}_{t}=\mathbb{E}^{\mathbb{Q}}[\psi \lvert\mathcal{F}_{t}]$ 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 $S_{t}(x)=\mathbb{E}^{\mathbb{Q}(x)}[\psi(x)\lvert\mathcal{F}_{t}]$ 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.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2563-2581.

Received: September 2017
Revised: July 2018
First available in Project Euclid: 4 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter 60H05: Stochastic integrals 91B51: Dynamic stochastic general equilibrium theory 91G99: None of the above, but in this section

Martingale representation property martingales stochastic integrals analytic fields endogenous completeness complete market equilibrium


Kramkov, Dmitry; Pulido, Sergio. Density of the set of probability measures with the martingale representation property. Ann. Probab. 47 (2019), no. 4, 2563--2581. doi:10.1214/18-AOP1321.

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  • [1] Anderson, R. M. and Raimondo, R. C. (2008). Equilibrium in continuous-time financial markets: Endogenously dynamically complete markets. Econometrica 76 841–907.
  • [2] Davis, M. and Obłój, J. (2008). Market completion using options. In Advances in Mathematics of Finance. Banach Center Publ. 83 49–60. Polish Acad. Sci. Inst. Math., Warsaw.
  • [3] German, D. (2011). Pricing in an equilibrium based model for a large investor. Math. Financ. Econ. 4 287–297.
  • [4] Hugonnier, J., Malamud, S. and Trubowitz, E. (2012). Endogenous completeness of diffusion driven equilibrium markets. Econometrica 80 1249–1270.
  • [5] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • [6] Kramkov, D. (2015). Existence of an endogenously complete equilibrium driven by a diffusion. Finance Stoch. 19 1–22.
  • [7] Kramkov, D. and Predoiu, S. (2014). Integral representation of martingales motivated by the problem of endogenous completeness in financial economics. Stochastic Process. Appl. 124 81–100.
  • [8] Kramkov, D. and Pulido, S. (2016). A system of quadratic BSDEs arising in a price impact model. Ann. Appl. Probab. 26 794–817.
  • [9] Kramkov, D. and Sîrbu, M. (2006). On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 16 1352–1384.
  • [10] Riedel, F. and Herzberg, F. (2013). Existence of financial equilibria in continuous time with potentially complete markets. J. Math. Econom. 49 398–404.
  • [11] Schwarz, D. C. (2017). Market completion with derivative securities. Finance Stoch. 21 263–284.