• Bernoulli
  • Volume 24, Number 3 (2018), 1613-1635.

Regularity of BSDEs with a convex constraint on the gains-process

Bruno Bouchard, Romuald Elie, and Ludovic Moreau

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


We consider the minimal super-solution of a backward stochastic differential equation with constraint on the gains-process. The terminal condition is given by a function of the terminal value of a forward stochastic differential equation. Under boundedness assumptions on the coefficients, we show that the first component of the solution is Lipschitz in space and $\frac{1}{2}$-Hölder in time with respect to the initial data of the forward process. Its path is continuous before the time horizon at which its left-limit is given by a face-lifted version of its natural boundary condition. This first component is actually equal to its own face-lift. We only use probabilistic arguments. In particular, our results can be extended to certain non-Markovian settings.

Article information

Bernoulli, Volume 24, Number 3 (2018), 1613-1635.

Received: September 2014
Revised: June 2015
First available in Project Euclid: 2 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

backward stochastic differential equation with a constraint regularity stability


Bouchard, Bruno; Elie, Romuald; Moreau, Ludovic. Regularity of BSDEs with a convex constraint on the gains-process. Bernoulli 24 (2018), no. 3, 1613--1635. doi:10.3150/16-BEJ806.

Export citation


  • [1] Bertsekas, D.P. and Shreve, S.E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering 139. New York–London: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers].
  • [2] Bouchard, B. and Nutz, M. (2016). Stochastic target games and dynamic programming via regularized viscosity solutions. Math. Oper. Res. 41 109–124.
  • [3] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
  • [4] Bouchard, B. and Vu, T.N. (2010). The obstacle version of the geometric dynamic programming principle: Application to the pricing of American options under constraints. Appl. Math. Optim. 61 235–265.
  • [5] Broadie, M., Cvitanić, J. and Soner, H.M. (1998). Optimal replication of contingent claims under portfolio constraints. Rev. Financ. Stud. 11 59–79.
  • [6] Cvitanić, J. and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3 652–681.
  • [7] Cvitanić, J., Karatzas, I. and Soner, H.M. (1998). Backward stochastic differential equations with constraints on the gains-process. Ann. Probab. 26 1522–1551.
  • [8] Cvitanić, J., Pham, H. and Touzi, N. (1999). Super-replication in stochastic volatility models under portfolio constraints. J. Appl. Probab. 36 523–545.
  • [9] Neveu, J. (1975). Discrete-Parameter Martingales, Revised ed. North-Holland Mathematical Library 10. Amsterdam–Oxford: North-Holland; New York: American Elsevier Publishing Co., Inc. Translated from the French by T.P. Speed.
  • [10] Pardoux, É. (1998). Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In Stochastic Analysis and Related Topics, VI (Geilo, 1996). Progress in Probability 42 79–127. Boston, MA: Birkhäuser.
  • [11] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Related Fields 113 473–499.
  • [12] Rockafellar, R.T. (1997). Convex Analysis. Princeton Landmarks in Mathematics and Physics 28. Princeton: Princeton University Press.
  • [13] Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.