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A class of backward doubly stochastic differential equations with discontinuous coefficients

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Abstract

In this work the existence of solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs) with coefficients left-Lipschitz in y (may be discontinuous) and Lipschitz in z is studied. Also, the associated comparison theorem is obtained.

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References

  1. Bally, V., Matoussi, A. Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab., 14(1): 125–164 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Duffie, D., Epstein, L. Stochastic differential utilities. Econometrica, 60(2): 354–439 (1992)

    Article  MathSciNet  Google Scholar 

  3. Hamadene, S., Lepeltier, J.-P. Backward equations, stochastic control and zero-sum stochastic differential games. Stoch. Stoch. Rep., 54(3–4): 221–231 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. 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(1): 230–239 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hu, M., Peng, S. On representation theorem of G-expectations and paths of G-Brownian motion. Acta Mathematicae Applicatae Sinica, 25(3): 539–546 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jia, G. A generalized existence theorem of BSDEs. C. R. Acad. Sci. Paris, Ser. I, 342(9): 685–688 (2006)

    Article  MATH  Google Scholar 

  7. Jia, G. On existence of backward stochastic differential equations with left-Lipschitz coefficient. Chin. J. of Contemp. Math., 28(4): 345–354 (2007) (in Chinese)

    Google Scholar 

  8. Jiang, L., Chen, Z. A result on the probability measures dominated by g-expectation. Acta Mathematicae Applicatae Sinica, 20(3): 1–6 (2004)

    MathSciNet  Google Scholar 

  9. Kim, K. L p estimates for SPDE with discontinuous coefficients in domains. Electron. J. Probab. 10(1): 1–20 (2005)

    MATH  MathSciNet  Google Scholar 

  10. Lin, Q. A class of backward doubly stochastic differential equations with non-Lipschitz coefficients. Statist. Probab. Lett., 79(20): 2223–2229 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lepeltier, J.P., Martin, J.S. Backward stochastic differential equations with continuous coefficients. Statist. Probab. Lett., 34(4): 425–430 (1997)

    Article  Google Scholar 

  12. Nualart, D., Pardoux, E. Stochastic calculus with anticipating integrands. Probab. Theory Related Fields, 78(4): 535–581 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. N’zi, M., Owo, J.M. Backward doubly stochastic differential equations with discontinuous coefficients. Statist. Probab. Lett., 79(7): 920–926 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pardoux, E., Peng, S. Adapted solution of a backward stochastic differential equation. Systems Control Letters, 14(1): 55–61 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pardoux, E., Peng, S. Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDE’s. Probab. Theory Related Fields, 98(2): 209–227 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Peng, S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep., 37(1–2): 61–74 (1991)

    MATH  Google Scholar 

  17. Peng, S. Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Mathematicae Applicatae Sinica, 20(2): 1–24 (2004)

    Google Scholar 

  18. Ren, Y., Lin, A., Hu, L. Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes. J. Comput. Appl. Math., 223(2): 901–907 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Shi, Y., Gu, Y., Liu, K. Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appal., 23(1): 97–110 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Yoo, H. L p-estimate for stochastic PDEs with discontinuous coefficients. Stoch. Anal. Appal., 17(4): 678–711 (1999)

    MathSciNet  Google Scholar 

  21. Zhang, Q., Zhao, H. Stationary solutions of SPDEs and infinite horizon BDSDEs. J. Funct. Anal., 252(1): 171–219 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China

    Qing-feng Zhu

  2. Institute for Financial Studies and School of Mathematics, Shandong University, Jinan, 250199, China

    Qing-feng Zhu & Yu-feng Shi

  3. School of Statistics, Shandong University of Finance and Economics, Jinan, 250014, China

    Yu-feng Shi

Authors
  1. Qing-feng Zhu
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  2. Yu-feng Shi
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Correspondence to Yu-feng Shi.

Additional information

Supported by the National Natural Science Foundation of China (Nos. 11371226, 11071145, 11301298, 11201268 and 11231005), Foundation for Innovative Research Groups of National Natural Science Foundation of China (No. 11221061), the 111 Project (No. B12023) and Natural Science Foundation of Shandong Province of China (ZR2012AQ013).

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Zhu, Qf., Shi, Yf. A class of backward doubly stochastic differential equations with discontinuous coefficients. Acta Math. Appl. Sin. Engl. Ser. 30, 965–976 (2014). https://doi.org/10.1007/s10255-011-0136-0

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  • DOI: https://doi.org/10.1007/s10255-011-0136-0

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