Bulletin of the Korean Mathematical Society (대한수학회보)

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Lp-SOLUTIONS FOR REFLECTED BSDES WITH TIME DELAYED GENERATORS

  • Zhou, Qing (School of Science Beijing University of Posts and Telecommunications)
  • Received : 2015.04.30
  • Published : 2016.05.31
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Abstract

In this paper, we establish the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations with time delayed generator (RBSDEs with time delayed generator, in short) in the case when the terminal value and the obstacle process are $L^p$-integrable with p ${\in}$]1, 2[ for a sufficiently small Lipschitz constant of the generator and the time horizon T.

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References

  1. S. Chen, $L^{p}$-solutions of one-dimensional backward stochastic differential equations with continuous coefficients, Stoch. Anal. Appl. 28 (2010), no. 5, 820-841. https://doi.org/10.1080/07362994.2010.503456
  2. S. Crepey and A. Moutoussi, Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison, Ann. Appl. Probab. 18 (2008), no. 5, 2041-2069. https://doi.org/10.1214/08-AAP517
  3. J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab. 24 (1996), no. 4, 2024-2056. https://doi.org/10.1214/aop/1041903216
  4. J. Cvitanic and J. Ma, Reflected forward-backward SDEs and obstacle problems with boundary conditions, J. Appl. Math. Stoch. Anal. 14 (2001), no. 2, 113-138. https://doi.org/10.1155/S1048953301000090
  5. C. Dellacherie and P. A. Meyer, Probabilites et Potentiel V-VIII, Hermann, Paris, 1980.
  6. L. Delong, Applications of time-delayed backward stochastic differential equations to pricing, Hedging and management of insurance and financial risks, 2010; PreprintarXiv:1005.4417.
  7. L. Delong and P. Imkeller, Backward stochastic differential equations with time delayed generators-results and counterexamples, Ann. Appl. Probab. 20 (2010), no. 4, 1512-1536. https://doi.org/10.1214/09-AAP663
  8. L. Delong and P. Imkeller, On Malliavin's differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures, Stochastic Process. Appl. 120 (2010), no. 9, 1748-1775. https://doi.org/10.1016/j.spa.2010.05.001
  9. N. El Karoui, C. Kapoudjian, E. Pardoux, P. Peng, and M. C. Quenez, Reflected solution of backward SDE's, and related obstacle problem for PDE's, Ann. Probab. 25 (1997), no. 2, 702-737. https://doi.org/10.1214/aop/1024404416
  10. N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1-71. https://doi.org/10.1111/1467-9965.00022
  11. N. El Karoui and M. C. Quenez, Non-linear pricing theory and backward stochastic differential equations, In W. J. Runggaldier (ed.), Financial Mathematics, Lecture Notes in Math. 1656, pp. 191-246, Springer-Verlag, Berlin, 1997.
  12. S. Fan and L. Jiang, $L^{p}$ solutions of finite and infinite time interval BSDEs with nonLipschitz coefficients, Stochastics 84 (2012), no. 4, 487-506. https://doi.org/10.1080/17442508.2011.615933
  13. S. Hamad'ene, BSDEs and risk sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stochastic Process. Appl. 107 (2003), no. 1, 145-169. https://doi.org/10.1016/S0304-4149(03)00059-0
  14. S. Hamad'ene and J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett. 24 (1995), no. 4, 259-263. https://doi.org/10.1016/0167-6911(94)00011-J
  15. S. Hamad'ene and J. P. Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stochastics Stochastics Rep. 54 (1995), no. 3-4, 221-231. https://doi.org/10.1080/17442509508834006
  16. S. Hamad'ene and J. P. Lepeltier, Reflected BSDEs and mixed game problem, Stochastic Process. Appl. 85 (2000), no. 2, 177-188. https://doi.org/10.1016/S0304-4149(99)00072-1
  17. S. Hamad'ene and A. Popier, $L^{p}$-solutions for reflected backward stochastic differential equations, Stoch. Dyn. 12 (2012), no. 2, 1150016, 35 pp.
  18. E. Pardoux, 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, Birkhauser Boston, Boston, MA, 1998.
  19. E. Pardoux, 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.
  20. E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55-61. https://doi.org/10.1016/0167-6911(90)90082-6
  21. E. Pardoux and S. Peng, 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.
  22. S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep. 37 (1991), no. 1-2, 61-74. https://doi.org/10.1080/17442509108833727
  23. G. Reis, A. Reveillac, and J. Zhang, FBSDEs with time delayed generators: $L^{p}$-solutions, differentiability, representation formulas and path regularity, Stochastic Process. Appl. 121 (2011), no. 9, 2114-2150. https://doi.org/10.1016/j.spa.2011.05.002
  24. Y. Ren and N. Xia, Generalized reflected BSDEs and an obstacle problem for PDEs with a nonlinear Neumann boundary condition, Stochastic Anal. Appl. 24 (2006), no. 5, 1013-1033. https://doi.org/10.1080/07362990600870454
  25. Q. Zhou and Y. Ren, Reflected backward stochastic differential equations with time delayed generators, Statist. Probab. Lett. 82 (2012), no. 5, 979-990. https://doi.org/10.1016/j.spl.2012.02.012