The Annals of Probability

Backward stochastic differential equations with reflection and Dynkin games

Jakša Cvitaniç and Ioannis Karatzas

Full-text: Open access


We establish existence and uniqueness results for adapted solutions of backward stochastic differential equations (BSDE's) with two reflecting barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez. Existence is proved first by solving a related pair of coupled optimal stopping problems, and then, under different conditions, via a penalization method. It is also shown that the solution coincides with the value of a certain Dynkin game, a stochastic game of optimal stopping. Moreover, the connection with the backward SDE enables us to provide a pathwise (deterministic) approach to the game.

Article information

Ann. Probab., Volume 24, Number 4 (1996), 2024-2056.

First available in Project Euclid: 6 January 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E05 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Backward SDE's reflecting barriers Dynkin games optimal stopping


Cvitaniç, Jakša; Karatzas, Ioannis. Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 (1996), no. 4, 2024--2056. doi:10.1214/aop/1041903216.

Export citation


  • Alario-Nazaret, M. (1982). Jeux de Dy nkin. Ph.D. dissertation, Univ. Franche-Comt´e, Besançon.
  • Alario-Nazaret, M., Lepeltier, J. P. and Marchal, B. (1982). Dy nkin games. Lecture Notes in Control and Inform. Sci. 43 23-42. Springer, Berlin.
  • Bensoussan, A. and Friedman, A. (1974). Non-linear variational inequalities and differential games with stopping times. J. Funct. Anal. 16 305-352.
  • Bismut, J. M. (1977). Sur un probl eme de Dy nkin. Z. Wahrsch. Verw. Gebiete 39 31-53.
  • Davis, M. H. A. and Karatzas, I. (1994). A deterministic approach to optimal stopping. In Probability, Statistics and Optimization (F. P. Kelly, ed.) 455-466. Wiley, New York.
  • Dellacherie, C. and Meyer, P. A. (1975), (1980). Probabilit´es et Potentiel. Chapts. I-IV, V-VIII. Hermann, Paris.
  • Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353-394.
  • Dunford, N. and Schwartz, J. T. (1963). Linear Operators. I: General Theory. Wiley, New York.
  • Dy nkin, E. B. and Yushkevich, A. A. (1968). Theorems and Problems in Markov Processes. Plenum Press, New York.
  • Ekeland, I. and Temam, R. (1976). Convex Analy sis and Variational Problems. North-Holland, Amsterdam.
  • El Karoui, N. (1981). Les aspects probabilistes du contr ole stochastique. Ecole d'Et´e de SaintFlour IX 1979. Lecture Notes in Math. 876 73-238. Springer, Berlin.
  • El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1995). Reflected solutions of backward SDE's and related obstacle problems for PDEs. Preprint.
  • Karatzas, I. (1993). Lecture Notes on Optimal Stopping Problems. Unpublished manuscript.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Lepeltier, J. P. and Maingueneau, M. A. (1984). Le jeu de Dy nkin en th´eorie g´en´erale sans l'hy poth ese de Mokobodski. Stochastics 13 25-44.
  • Morimoto, H. (1984). Dy nkin games and martingale methods. Stochastics 13 213-228.
  • Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.
  • Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Sy stems Control Lett. 14 55-61.
  • Stettner, L. (1982). Zero-sum Markov games with stopping and impulsive strategies. Appl. Math. Optim. 9 1-24.