The Annals of Probability

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

Dmitry Kramkov and Sergio Pulido

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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.

Export citation


  • [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.