×

zbMATH — the first resource for mathematics

Reflected generalized backward doubly SDEs driven by Lévy processes and applications. (English) Zbl 1259.60062
Summary: We study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with Lévy process with one continuous barrier. Under uniformly Lipschitz coefficients, we prove an existence and uniqueness result by means of the penalization method and the fixed-point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs) with a nonlinear Neumann boundary condition.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Dellacherie, C., Meyer, P.: Probabilities and Potential. North-Holland Mathematics Studies, vol. 29. North-Holland, Amsterdam (1978). viii+189 pp. · Zbl 0494.60001
[2] El Otmani, M.: Generalized BSDE driven by a Lévy process. J. Appl. Math. Stoch. Anal. (2006). Art. ID 85407, 25 pp. · Zbl 1147.60319
[3] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenz, M.C.: Reflected solution of backward SDE’s, and related obstacle problem for PDE’s. Ann. Probab. 25(2), 702–737 (1997) · Zbl 0899.60047 · doi:10.1214/aop/1024404416
[4] El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7(1), 1–71 (1997) · Zbl 0884.90035
[5] Hamadène, S.: Reflected BSDE’s with discontinuous barrier and application. Stoch. Stoch. Rep. 74(3–4), 571–596 (2002) · Zbl 1015.60057 · doi:10.1080/1045112021000036545
[6] Hamadène, S., Lepeltier, J.-P.: Zero-sum stochastic differential games and backward equations. Syst. Control Lett. 24(4), 259–263 (1995) · Zbl 0877.93125 · doi:10.1016/0167-6911(94)00011-J
[7] Hamadène, S., Ouknine, Y.: Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab. 8(2) (2003), 20 pp. · Zbl 1015.60051
[8] Hamadène, S., Lepeltier, J.-P., Matoussi, A.: Double barrier backward SDEs with continuous coefficient. In: Backward Stochastic Differential Equations, Paris, 1995–1996. Pitman Res. Notes Math. Ser., vol. 364, pp. 161–175. Longman, Harlow (1997) · Zbl 0887.60065
[9] Hu, L., Ren, Y.: Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes. J. Comput. Appl. Math. 229, 230–239 (2009) · Zbl 1173.60023 · doi:10.1016/j.cam.2008.10.027
[10] Matoussi, A.: Reflected solutions of backward stochastic differential equations with continuous coefficient. Stat. Probab. Lett. 34(4), 347–354 (1997) · Zbl 0882.60057 · doi:10.1016/S0167-7152(96)00202-7
[11] Menaldi, J., Robin, M.: Reflected diffusion processes with jumps. Ann. Probab. 13(2), 319–341 (1985) · Zbl 0565.60065 · doi:10.1214/aop/1176992994
[12] Nualart, D., Schoutens, W.: Chaotic and predictable representations for Lévy processes. Stoch. Process. Appl. 90(1), 109–122 (2000) · Zbl 1047.60088 · doi:10.1016/S0304-4149(00)00035-1
[13] Nualart, D., Schoutens, W.: Backward stochastic differential equations and Feynman–Kac formula for Lévy processes, with applications in finance. Bernoulli 7(5), 761–776 (2001) · Zbl 0991.60045 · doi:10.2307/3318541
[14] Pardoux, E., Peng, S.: Adapted solution of backward stochastic differential equation. Syst. Control Lett. 4(1), 55–61 (1990) · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[15] Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications, Charlotte, NC, 1991. Lecture Notes in Control and Inform. Sci., vol. 176, pp. 200–217. Springer, Berlin (1992) · Zbl 0766.60079
[16] Pardoux, E., Peng, S.: Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat. Fields 98(2), 209–227 (1994) · Zbl 0792.60050 · doi:10.1007/BF01192514
[17] Pardoux, E., Zhang, S.: BSDEs Generalized nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields 110(4), 535–558 (1998) · Zbl 0909.60046 · doi:10.1007/s004400050158
[18] Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Stochastic Modeling and Applied Probability. Springer, Berlin (2005), Version 2.1
[19] Qing, Z.: On comparison theorem and solutions of BSDEs for Lévy processes. Acta Math. Appl. Sin., Engl. Ser. 23(3), 513–522 (2007) · Zbl 1131.60059 · doi:10.1007/s10255-007-0391-2
[20] Ren, Y., El Otmani, M.: Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition. J. Comput. Appl. Math. (2009). doi: 10.1016/j.cam.2009.09.037 · Zbl 1217.60047
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.