Electronic Journal of Probability

Stability and Markov property of forward backward minimal supersolutions

Samuel Drapeau and Christoph Mainberger

Full-text: Open access


We show stability and locality of the minimal supersolution of a forward backward stochastic differential equation with respect to the underlying forward process under weak assumptions on the generator. The forward process appears both in the generator and the terminal condition. Painlevé-Kuratowski and Convex Epi-convergence are used to establish the stability. For Markovian forward processes the minimal supersolution is shown to have the Markov property. Furthermore, it is related to a time-shifted problem and identified as the unique minimal viscosity supersolution of a corresponding PDE.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 41, 15 pp.

Received: 30 April 2015
Accepted: 17 May 2016
First available in Project Euclid: 17 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 35D40: Viscosity solutions

supersolutions of backward stochastic differential equations stability FBSDEs Markov property viscosity supersolutions

Creative Commons Attribution 4.0 International License.


Drapeau, Samuel; Mainberger, Christoph. Stability and Markov property of forward backward minimal supersolutions. Electron. J. Probab. 21 (2016), paper no. 41, 15 pp. doi:10.1214/16-EJP4276. https://projecteuclid.org/euclid.ejp/1466166072

Export citation


  • [1] J.P. Aubin and H. Frankowska. Set-Valued Analysis. Modern Birkhäuser Classics. Birkhäuser, 2009.
  • [2] Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., 27(1):1–67, July 1992.
  • [3] Freddy Delbaen, Ying Hu, and Xiaobo Bao. Backward SDEs with Superquadratic Growth. Probability Theory and Related Fields, 150(1-2):145–192, 2011.
  • [4] Samuel Drapeau, Gregor Heyne, and Michael Kupper. Minimal Supersolutions of Convex BSDEs. Annals of Probability, 41(6):3973–4001, 2013.
  • [5] Samuel Drapeau, Michael Kupper, Emanuela Rosazza Gianin, and Ludovic Tangpi. Dual Representation of Minimal Supersolutions of Convex BSDEs. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 52(2):868–887, 2016.
  • [6] Nicole El Karoui, Shige Peng, and Marie Claire Quenez. Backward Stochastic Differential Equations in Finance. Mathematical Finance, 7(1):1–71, 1997.
  • [7] Henner Gerdes, Gregor Heyne, and Michael Kupper. Stability of closed convex hulls and minimal supersolutions of convex BSDEs. Preprint, 2013.
  • [8] Gregor Heyne, Michael Kupper, and Christoph Mainberger. Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 50(2):524–538, 2014.
  • [9] Andreas Löhne and Constantin Zălinescu. On convergence of closed convex sets. Journal of Mathematical Analysis and Applications, 319(2):617–634, 2006.
  • [10] Etienne Pardoux and Shige Peng. Adapted Solution of a Backward Stochastic Differential Equation. System & Control Letters, 14(1):55–61, 1990.
  • [11] Etienne Pardoux and Shige Peng. Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations. Lectures Notes in CIS, 176:200–217, 1992.
  • [12] Etienne Pardoux and Aurel Răşcanu. Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Springer International Publishing, 2014.
  • [13] Shige Peng. Probabilistic Interpretation for Systems of Quasilinear Parabolic Partial Differential Equations. Stochastics, 37:61–74, 1992.
  • [14] Ralph Tyrrel Rockafellar. Convex Analysis. Princeton University Press. Princeton, New Jerzey, 1970.
  • [15] Ralph Tyrrel Rockafellar and Roger J-B Wets. Variational Analysis. Springer, Berlin, New York, 1998.
  • [16] Nizar Touzi. Optimal Stochastic Control, Stochastic Target Problems, and Backward SDEs. Fields Institute Monographs. Springer, 2012.