African Diaspora Journal of Mathematics

Reflected Generalized BSDEs with Random Time and Applications

A. Aman, A. Elouaflin, and M. N’Zi

Full-text: Open access


In this paper, we aim to study solutions of reflected generalized BSDEs, involving the integral with respect to a continuous process, which is the local time of the diffusion on the boundary. We consider both a finite random terminal and a infinite horizon. In both case, we establish an existence and uniqueness result. As application, we give a characterization of an American pricing option in infinite horizon; and we also give a probabilistic formula for the viscosity solution of an obstacle problem for elliptic PDEs with a nonlinear Neumann boundary condition.

Article information

Afr. Diaspora J. Math. (N.S.), Volume 14, Number 1 (2012), 83-105.

First available in Project Euclid: 18 July 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H20, 60H30, 60H99

American option pricing elliptic PDEs generalized backward stochastic differential equations Neumann boundary condition viscosity solution


Aman, A.; Elouaflin, A.; N’Zi, M. Reflected Generalized BSDEs with Random Time and Applications. Afr. Diaspora J. Math. (N.S.) 14 (2012), no. 1, 83--105.

Export citation


  • Crandall, M.; Ichii, H. and Lions, P. L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67.
  • Cvitanic, J. and Ma, J., Hedging option for a large investor and Forward-Backward SDE's. Ann. Appl. Probab. 6 (1996), no. 2, 370-398.
  • Cvitanic J. and Ma J., Reflected forward-backward SDEs and obstacle problems with boundary conditions. J. Appl. Math. Stochastic Anal. 14 (2001), no. 2, 113-138.
  • Dellacherie, C. and Meyer, P. A. Probabilités et potentiel. (French) Chapitres I a IV. Édition entièrement refondue. Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV. Actualités Scientifiques et Industrielles, No. 1372. Hermann, Paris, 1975. x+291 pp.
  • El Karoui N., Les aspects probabilistes du contrôle stochastique. (French) [The probabilistic aspects of stochastic control] Ninth Saint Flour Probability Summer School–1979 (Saint Flour, 1979), pp. 73-238, Lecture Notes in Math., 876, Springer, Berlin-New York, 1981.
  • El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equation in finance. Math. Finance. 7 (1997), no. 1, 1-71.
  • El Karoui, N.; Kapoudjian, C.; Pardoux E.; Peng, S. and Quenz, M. C., Reflected solution of backward SDE's and related obstacle problem for PDE's, Ann. Probab. 25 (1997), no. 2, 702-737.
  • Hamadène, S. and Lepeltier, J. P., Zero-sum stochastic games and BSDEs, Systems Control Lett. 24 (1995), no. 4, 259-263.
  • Hamadène, S. and Lepeltier, J. P., Reflected BSDes and mixed game problem, Stochastic Process. Appl 85 (2000), no. 2, 177-188.
  • Hamadène, S.; Lepeltier J. P. and Matoussi, A., Double barrier backward SDEs with continuous coefficient. Backward stochastic differential equations (Paris, 1995-1996), 161-175, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997.
  • Hamadène S., Ouknine Y., Reflected Backward Stochastic Differential Equations with jumps and random obstacle. Electron. J. Probab. 8 (2003), no. 2, 20 pp.
  • Lions, P. L. and Sznitman, A. S., Stochastic differential equations with reflecting boundary conditions. Comm. Pure and Appl. Math. 37 (1984), no. 4, 511-537.
  • N'Zi, M. et Ouknine, Y., Multivalued backward stochastic differential equations.Probab. Math. Statist. 17 (1997), no. 2, 259-275.
  • Pardoux, E, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order Stochastic analysis and related topics, VI (Geilo, 1996), 79-127, Progr. Probab., 42, Birkhäuser Boston, Boston, MA, 1998.
  • Pardoux, E., BSDEs, weak convergence and homogenization of semilinear PDEs. Nonlinear analysis, differential equations and control (Montreal, QC, 1998), 503-549, NATO Sci. Ser. C Math. Phys. Sci., 528, Kluwer Acad. Publ., Dordrecht, 1999.
  • Pardoux, E. and Peng S., Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications (Charlotte, NC, 1991), 200-217, Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992.
  • Pardoux, E., Zhang S., generalized BSDEs and nonlinear Neumann boundary value problems. Proba. Theory and Related Fields 110 (1998), no. 4, 535-558.
  • Ren, Y. and Xia, N. Generalized reflected BSDE and obstacle problem for PDE with nonlinear Neumann boundary condition. Stoch. Anal. Appl. 24 (2006), no. 5, 1013-1033.
  • Ren, Y. and Hu, L., Reflected backward stochastic differential equations driven by Lévy processes. Statist. Probab. Lett. 77 (2007), no. 15, 1559-1566.
  • Saisho, Y., Stochastic differential equation for multidimensional domains with reflecting boundary. Probab. Theory Related Fields 74 (1987), no. 3, 455-477.