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Ukrainian Mathematical Journal

, Volume 64, Issue 6, pp 857–874 | Cite as

A comonotonic theorem for backward stochastic differential equations in L p and its applications

  • Z.-J. Zong
Article

We study backward stochastic differential equations (BSDE) under weak assumptions on the data. We obtain a comonotonic theorem for BSDE in L p ; 1 < p ≤ 2: As applications of this theorem, we study the relation between Choquet expectations and minimax expectations and the relation between Choquet expectations and generalized Peng’s g-expectations. These results generalize the well-known results of Chen et al.

Keywords

Stochastic Differential Equation Backward Stochastic Differential Equation Stochastic Differential Game Null Subset Backward Stochastic Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Stoica, “L p solutions of backward stochastic differential equations,” Stochast. Proc. Appl., 108, 109–129 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Z. Chen, T. Chen, and M. Davison, “Choquet expectation and Peng’s g-expectation,” Ann. Probab., 33, No. 3, 1179–1199 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Z. Chen and L. Epstein, “Ambiguity, risk, and asset returns in continuous time,” Econometrica, 70, No. 4, 1403–1443 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Z. Chen, R. Kulperger, and G. Wei, “A comonotonic theorem for BSDE,” Stochast. Proc. Appl., 115, 41–54 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Z. Chen and R. Kulperger, “Minimax pricing and Choquet pricing,” Insurance: Math. Econ., 38, No. 3, 518–528 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    G. Choquet, “Theory of capacities,” Ann. Inst. Fourier, 5, 131–195 (1953).MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. W. R. Darling, “Constructing gamma martingales with prescribed limits, using BSDE,” Ann. Probab., 23, 1234–1261 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    N. El Karoui, S. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Math. Finance, 7, No. 1, 1–71 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Hamadene and J. Lepeltier, “Zero-sum stochastic differential games and BSDE,” Stochast. Stochast. Repts, 54, 221–231 (1995).MathSciNetzbMATHGoogle Scholar
  10. 10.
    F. Hu and Z. Chen, “Generalized Peng’s g-expectation and related properties,” Statist. Probab. Lett., 80, 191–195 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Hu, “On the integral representation of g-expectations,” Compt. Rend. Math., 348, 571–574 (2010).zbMATHGoogle Scholar
  12. 12.
    Y. Hu, “Probabilistic interpretation for systems of quasielliptic PDE with Neumann boundary conditions,” Stochast. Proc. Appl., 48, 107–121 (1993).CrossRefzbMATHGoogle Scholar
  13. 13.
    J. Ma, J. Protter, and J. Yong, “Solving forward-backward stochastic differential equations—a four-step scheme,” Probab. Theory Relat. Fields, 98, 339–359 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E. Pardoux and S. Peng, “Adapted solution of a backward stochastic differential equation,” Syst. Control Lett., 14, No. 1, 55–61 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    E. Pardoux and S. Zhang, “Generalized BSDE and nonlinear Neumann boundary-value problems,” Probab. Theory Relat. Fields, 110, 535–558 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Peng, “Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,” Stochast. Stochast. Rep., 37, 61–74 (1991).zbMATHGoogle Scholar
  17. 17.
    S. Peng, “Backward SDE and related g-expectation,” Pitman Res. Notes in Math. Ser., Vol. 364, N. El Karoui and L. Mazliak (editors), Backward Stochastic Differential Equations, Longman, Harlow (1997), pp. 141–159.Google Scholar
  18. 18.
    S. Peng, “Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type,” Probab. Theory Relat. Fields, 113, 473–499 (1999).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Z.-J. Zong
    • 1
  1. 1.School of Mathematical SciencesQufu Normal UniversityShandongChina

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