BSDEs with two reflecting barriers driven by a Brownian motion and Poisson noise and related Dynkin game

Said Hamadene (Universite du Maine, France)
Mohammed Hassani (Universite du Maine, France)

Abstract


In this paper we study BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson process. We show the existence and uniqueness of $local$ and global solutions. As an application we solve the related zero-sum Dynkin game.

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Pages: 121--145

Publication Date: February 15, 2006

DOI: 10.1214/EJP.v11-303

References

  1. G.Barles, R.Buchdahn, E.Pardoux, BSDE's and integral-partial differential equations, Stochastics 60, 57-83, (1997) Math. Review 98d:60110
  2. C.Dellacherie, P.A.Meyer, Probabilits et Potentiel, Chap. I-IV, Hermann, Paris (1975) Math. Review MR0488194
  3. C.Dellacherie, P.A.Meyer, Probabilits et Potentiel. Chap. V-VIII, Hermann, Paris (1980) Math. Review 82b:60001
  4. J.Cvitanic, I.Karatzas, Backward SDEs with reflection and Dynkin games, Annals of Probability 24 (4), pp. 2024-2056 (1996) MR1415239
  5. J.Cvitanic, J.Ma, Reflected Forward-Backward SDEs and Obstacle Problems with Boundary Conditions, Journal of Applied Math. and Stochastic Anal., 14:2 (2001), 113-138 MR1398050
  6. N.El-Karoui, C.Kapoudjian, E.Pardoux, S.Peng, M.C.Quenez, Reflected solutions of backward SDEs and related obstacle problems for PDEs, Annals of Probability 25 (2) (1997), pp.702-737 Math. Review 98k:60096
  7. N.El-Karoui, Les aspects probabilistes du contrle stochastique, in Ecole d't de Saint-Flour. Lecture Notes in Mathematics 876, 73-238, Springer Verlag Berlin (1980) Math. Review 83c:93062
  8. N.El-Karoui, E.Pardoux, M.-C.Quenez, Reflected backward SDEs and American options, in: Numerical Methods in Finance (L.Robers and D. Talay eds.), Cambridge U. P., 215-231 (1997) MR1470516
  9. S.Hamadne, Mixed Zero-sum differential game and American game options, to appear in SIAM JCO (2006)
  10. S.Hamadne, J.-P.Lepeltier, Reflected BSDEs and mixed game problem, Stochastic Processes and their Applications 85 (2000) p. 177-188 Math. Review 2001g:91025
  11. S.Hamadne, J.-P.Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stochastic and Stochastic Reports 54 (1995), 221-231 MR1382117
  12. S.Hamadne, M.Jeanblanc, On the stopping and starting problem: application in reversible investment, to appear in Mathematics of Operation Research (2006)
  13. S.Hamadne, Y.Ouknine, Reflected backward stochastic differential equation with jumps and random obstacle, Electronic Journal of Probability. Vol. 8 (2003) Paper no.2, pages 1-20, MR1121940
  14. Y.Kifer, Game Options, Finance and Stochastics 4 (2000), pp. 443-463 MR1779588
  15. J.P.Lepeltier, M.A. Maingueneau, Le jeux de Dynkin en thorie gnrale sans l'hypothse de Mokobodski, Stochastics (1984), vol.13, p.25-44 MR0752475
  16. X.Li , S.Tang, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM JCO, 32, pp. 1447-1475 (1994) MR1288257
  17. E. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, in: F. Clarke and R. Stern (eds), Nonlin. Anal., Dif. Equa. and Control, 503-549 (1999), Kluwer Acad. Publi., Netherlands MR1695013
  18. E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Systems and Control Letters 14, pp. 55-61 (1990) Math. Review 91e:60171
  19. E. Pardoux, S. Peng, Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, in: Stochastic Differential Equations and their Applications (B. Rozovskii and R. Sowers, eds.), Lect. Not. Cont. Inf. Sci., vol.176, Springer, 200-217 (1992) MR1176785
  20. D.Revuz, M.Yor, Continuous Martingales and Brownian Motion, Springer, Berlin (1991) Math. Review 92d:60053
  21. N.Touzi, N.Vieille, Continuous-time Dynkin games with Mixed Strategies, SIAM JCO vol. 41 (4) (2002), pp.1073-1088. MR1972503
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