Advertisement

SeMA Journal

, Volume 76, Issue 1, pp 37–63 | Cite as

\(\mathbb {L}^2\)-solutions for reflected BSDEs with jumps under monotonicity and general growth conditions: a penalization method

  • Imade FakhouriEmail author
  • Youssef Ouknine
Article
  • 21 Downloads

Abstract

In this paper, we study generalized reflected backward stochastic differential equations with a càdlàg barrier, in a general filtration that supports a Brownian motion and an independent Poisson random measure. We give necessary and sufficient conditions for existence and uniqueness of \(\mathbb {L}^2\)-solutions for equations with generators monotone in y. We also prove that the solutions can be approximated via the penalization method. Furthermore, a comparison theorem is provided for such equations.

Keywords

Reflected backward stochastic differential equation General filtration Jumps Penalization method 

Mathematics Subject Classification

60H10 60H20 

References

  1. 1.
    Aazizi, S., El Mellali, T., Fakhouri, I., Ouknine, Y.: Optimal switching problem and related system of BSDEs with left-Lipschitz coefficients and mixed reflections. Stat. Probab. Lett. 137, 70–78 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aazizi, S., Fakhouri, I.: Optimal switching problem and system of reflected multi-dimensional FBSDEs with random terminal time. Bull. Sci. Math. 137(4), 523–540 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aman, A.: \({\mathbb{L}}^p\)-solution of reflected generalized BSDEs with non-Lipschitz coefficients. Random Oper. Stoch. Equ. 17, 201–219 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Int. J. Probab. Stoch. Process. 60, 57–83 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bouchard, B., Possamai, D., Tan, X., Zhou, C.: A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations. Ann. Inst. H. Poincaré Probab. Stat. 54(1), 154–172 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cvitanic, J., Karatzas, I.: Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24, 2024–2056 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eddahbi, M., Fakhouri, I., Ouknine, Y.: \({\mathbb{L}}^p(p\ge 2)\) solutions of generalized BSDEs with jumps and monotone generator in a general filtration. Modern Stoch. Theory Appl. 4(1), 25–63 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    El Asri, B., Fakhouri, I.: Viscosity solutions for a system of PDEs and optimal switching. IMA J. Math. Control Inf. 34(3), 937–960 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.-C.: Reflected solutions of backward SDE’s and related obstacle problems for PDE’s. Ann. Probab. 25, 702–737 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    El Karoui, N., Pardoux, E., Quenez, M.-C.: Reflected backward SDEs and American options. In: Rogers, L.C.G., Talay, D. (eds.) Numer. Methods Financ., pp. 215–231. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  11. 11.
    Essaky, E.-H.: Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. des Sci. Mathématiques 132, 690–710 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fan, S.-J.: Existence uniqueness and approximation for \({\mathbb{L}}^p\)-solutions of reflected BSDEs under weaker assumptions (2015). arXiv:1510.08587
  13. 13.
    Grigorova, M., Imkeller, P., Offen, E., Ouknine, Y., Quenez, M.-C.: Reflected BSDEs when the obstacle is not right-continuous and optimal stopping. Ann. Appl. Probab. 27(5), 3153–3188 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grigorova, M., Imkeller, P., Ouknine, Y., Quenez, M.-C.: Optimal stopping with f-expectations: the irregular case. Ann. Appl. Probab. (2016). arXiv:1611.09179
  15. 15.
    Hamadène, S.: Reflected BSDE’s with discontinuous barrier and application. Stoch. Int. J. Probab. Stoch. Process. 74, 571–596 (2002)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hamadène, S., Jeanblanc, M.: On the starting and stopping problem: application in reversible investments. Math. Oper. Res. 32, 182–192 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hamadène, S., Lepeltier, J.-P.: Reflected BSDEs and mixed game problem. Stoch. Process. Appl. 85, 177–188 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hamadène, S., Ouknine, Y.: Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab. 8, 1–20 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hamadène, S., Ouknine, Y.: Reflected backward SDEs with general jumps. Theory Probab. Appl. 60, 263–280 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hamadène, S., Popier, A.: \({\mathbb{L}}^p\)-solutions for reflected backward stochastic equations. Stochastics Dyn. 12(2), 1150016 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hamadène, S., Zhang, J.: Switching problem and related system of reflected backward SDEs. Stoch. Process. Appl. 120, 403–426 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hu, Y., Tang, S.: Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Relat. Fields 147, 89–121 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, Grundlehren der mathematischen Wissenschaften, vol. 288. Springer, Berlin (2003)Google Scholar
  24. 24.
    Klimsiak, T.: Reflected BSDEs with monotone generator. Electron. J. Probab. 17(107), 1–25 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Klimsiak, T.: BSDEs with monotone generator and two irregular reflecting barriers. Bull. des Sci. Mathématiques 137, 268–321 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Klimsiak, T.: Reflected BSDEs on filtered probability spaces. Stoch. Process. Appl. 125, 4204–4241 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Klimsiak, T., Rzymowski, M., Słomiński, L.: Reflected BSDEs with regulated trajectories (2016). arXiv:1608.08926
  28. 28.
    Kruse, T., Popier, A.: BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stoch. Int. J. Probab. Stoch. Process. 88(4), 1–49 (2015)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lepeltier, J.-P., Matoussi, A., Xu, M.: Reflected backward stochastic differential equations under monotonicity and general increasing growth conditions. Adv. Appl. Probab. 37, 134–159 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lepeltier, J.-P., Xu, M.: Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier. Statist. Probab. Lett. 75, 58–66 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Peng, S.: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyers type. Probab. Theory Relat. Fields 113, 473–499 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Peng, S., Xu, M.: The smallest g-supermartingale and reflected BSDE with single and double L2 obstacles. Ann. l’Institut Henri Poincare Probab. Stat. 41, 605–630 (2005)CrossRefzbMATHGoogle Scholar
  34. 34.
    Quenez, M.-C., Sulem, A.: BSDEs with jumps optimization and applications to dynamic risk measures. Stoch. Process. Appl. 123, 3328–3357 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Quenez, M.-C., Sulem, A.: Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps. Stoch. Process. Appl. 124, 3031–3054 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116, 1358–1376 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Rozkosz, A., Słomiński, L.: \({\mathbb{L}}^p\)-solutions of reflected BSDEs under monotonicity condition. Stoch. Process. Appl. 122, 3875–3900 (2012)CrossRefzbMATHGoogle Scholar
  38. 38.
    Xu, M.: Reflected backward SDEs with two barriers under monotonicity and general increasing conditions. J. Theor. Probab. 20, 1005–1039 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences SemlaliaCadi Ayyad UniversityMarrakeshMorocco
  2. 2.The Hassan II Academy of Sciences and TechnologyRabatMorocco

Personalised recommendations