×

Backward stochastic differential equations with reflection and Dynkin games. (English) Zbl 0876.60031

The authors study backward stochastic differential equations with reflection (RBSDE’s) on two stochastic barriers (upper and lower boundary processes) in relation both with Dynkin games and coupled optimal stopping problems. The uniqueness of the solution of an RBSDE is proved using that any such solution is the (unique) value of a Dynkin game. The existence of a solution of this RBSDE is proved using that a pair of coupled optimal stopping problems has a solution. The authors present also an alternative method to prove existence of a solution of the RBSDE using a more classical penalization method. In the last section the authors study a pathwise approach to the Dynkin game.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
91A60 Probabilistic games; gambling
60G40 Stopping times; optimal stopping problems; gambling theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alario-Nazaret, M. (1982). Jeux de Dy nkin. Ph.D. dissertation, Univ. Franche-Comté, Besançon.
[2] 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.
[3] Bensoussan, A. and Friedman, A. (1974). Non-linear variational inequalities and differential games with stopping times. J. Funct. Anal. 16 305-352. · Zbl 0297.90120
[4] Bismut, J. M. (1977). Sur un probl eme de Dy nkin. Z. Wahrsch. Verw. Gebiete 39 31-53. · Zbl 0336.60069
[5] 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. · Zbl 0855.60041
[6] Dellacherie, C. and Meyer, P. A. (1975), (1980). Probabilités et Potentiel. Chapts. I-IV, V-VIII. Hermann, Paris.
[7] Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353-394. JSTOR: · Zbl 0768.90006
[8] Dunford, N. and Schwartz, J. T. (1963). Linear Operators. I: General Theory. Wiley, New York. · Zbl 0128.34803
[9] Dy nkin, E. B. and Yushkevich, A. A. (1968). Theorems and Problems in Markov Processes. Plenum Press, New York.
[10] Ekeland, I. and Temam, R. (1976). Convex Analy sis and Variational Problems. North-Holland, Amsterdam. · Zbl 0322.90046
[11] El Karoui, N. (1981). Les aspects probabilistes du contr ole stochastique. Ecole d’Eté de SaintFlour IX 1979. Lecture Notes in Math. 876 73-238. Springer, Berlin. · Zbl 0472.60002
[12] 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. · Zbl 0899.60047
[13] Karatzas, I. (1993). Lecture Notes on Optimal Stopping Problems. Unpublished manuscript.
[14] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York. · Zbl 0734.60060
[15] Lepeltier, J. P. and Maingueneau, M. A. (1984). Le jeu de Dy nkin en théorie générale sans l’hy poth ese de Mokobodski. Stochastics 13 25-44. · Zbl 0541.60041
[16] Morimoto, H. (1984). Dy nkin games and martingale methods. Stochastics 13 213-228. · Zbl 0569.60050
[17] Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam. · Zbl 0345.60026
[18] Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Sy stems Control Lett. 14 55-61. · Zbl 0692.93064
[19] Stettner, L. (1982). Zero-sum Markov games with stopping and impulsive strategies. Appl. Math. Optim. 9 1-24. · Zbl 0524.60047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.