Geodesic
Anticipated backward doubly stochastic differential equations driven by Teugels martingales with or without reflecting barrier
Daghestan Electronic Mathematical Reports, Tome 13 (2020), pp. 1-21.

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We are interested in this paper on reflected anticipated backward doubly stochastic differential equations (RABDSDEs) driven by teugels martingales associated with Levy process. We obtain the existence and uniqueness of solutions to these equations by means of the fixed-point theorem where the coefficients of these BDSDEs depend on the future and present value of the solution $(Y, Z)$. We also show the comparison theorem for a special class of RABDSDEs under some slight stronger conditions. The novelty of our result lies in the fact that we allow the time interval to be infinite.
Keywords: anticipated backward doubly stochastic differential equations, random Levy measure, comparison theorem, predictable representation, Teugels martingales. 
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     title = {Anticipated backward doubly stochastic differential equations driven by {Teugels} martingales with or without reflecting barrier},
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M. Saouli; B. Mansouri. Anticipated backward doubly stochastic differential equations driven by Teugels martingales with or without reflecting barrier. Daghestan Electronic Mathematical Reports, Tome 13 (2020), pp. 1-21. http://geodesic.mathdoc.fr/item/DEMR_2020_13_a0/

[1] Bahlali K., Hassani M., Mansouri B., Mrhardy N., “One barrier reflected backward doubly stochastic differential equations with continuous generator”, Comptes Rendus Mathematique, 347:19 (2009), 59–68 | DOI | MR

[2] Bertoin J., Levy Processes, Cambridge University Press, Cambridge–London, 1996 | MR | Zbl

[3] Nualart D. and Schoutens W., “Chaotic and predictable representation for Levy process”, Stochastic Process. Appl., 90 (2000), 109–122 | DOI | MR | Zbl

[4] Nualart D. and Schoutens W., “Backward stochastic differential equations and Feynman–Kac formula for Levy processes, with applications in finance”, Bernoulli., 5 (2001), 761–776 | DOI | MR | Zbl

[5] Pardoux E., Peng S., “Adapted solution of a backward stochastic differential equation”, Systems Control Lett., 4 (1990), 55–61 | DOI | MR

[6] Pardoux E., Peng S., “Backward doubly stochastic differential equations and systems of quasili near SPDEs”, Probability Theory and Related Fields, 98 (1994), 209–227 | DOI | MR | Zbl

[7] Peng S. and Yang Z., “Anticipated backward stochastic differential equations”, Ann. Probab., 37 (2009), 877–902 | DOI | MR | Zbl

[8] Ren Y., Lin A., Hu L., “Stochastic PDIEs and backward doubly stochastic differential equations driven by Levy processes”, Computational and Applied Mathematics, 223 (2009), 901–907 | DOI | MR | Zbl

[9] Ren Y., “Reflected backward doubly stochastic differential equations driven by Levy processes”, C.R. Acad. Sci. Paris Ser. I., 384 (2010), 439–444 | DOI | MR

[10] Xu X., “Anticipated backward doubly stochastic differential equations”, Appl. Math. Comput., 22 (2013). 53–62 | MR

[11] Zong G., “Anticipated backward stochastic differential equations driven by the Teugels martingales”, Mathematical Analysis and Applications, 412:2 (2014), 989–997 | DOI | MR | Zbl