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L p Theory for Super-Parabolic Backward Stochastic Partial Differential Equations in the Whole Space

Abstract

This paper is concerned with semi-linear backward stochastic partial differential equations (BSPDEs for short) of super-parabolic type. An L p-theory is given for the Cauchy problem of BSPDEs, separately for the case of p∈(1,2] and for the case of p∈(2,∞). A comparison theorem is also addressed.

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References

  1. Bensoussan, A.: Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions. Stochastics 9, 169–222 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bismut, J.: Linear quadradic optimal stochastic control with random coefficients. SIAM J. Control Optim. 14, 414–444 (1976)

    Article  MathSciNet  Google Scholar 

  3. Bismut, J.: Contrôl des systèmes linéares quadratiques. In: Applications de L’intégrale Stochastique, Séminaire de Probabilité XII. Lecture Notes in Mathematics, vol. 649, pp. 180–264. Springer, Berlin (1978)

    Google Scholar 

  4. Bismut, J.: An introductory approach to duality in optimal stochastic control. SIAM Rev. 20, 62–78 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Briand, P., Delyon, B., Hu, Y., Pardoux, E., Stoica, L.: Lp solutions of backward stochastic differential equations. Stoch. Process. Appl. 108, 604–618 (2003)

    Article  MathSciNet  Google Scholar 

  6. Delbaen, F., Tang, S.: Harmonic analysis of stochastic equations and backward stochastic differential equations. Probab. Theory Relat. Fields 146, 291–336 (2010)

    Article  MathSciNet  Google Scholar 

  7. Dokuchaev, N.: Backward parabolic Itô equations and second fundamental inequality (2010). arXiv:math/0606595v3

  8. Du, K.: On semi-linear degenerate backward stochastic PDEs in Rd (2011). Preprint

  9. Du, K., Chen, S.: Semi-linear backward stochastic PDEs with quadratic growth in general domains (2011). Preprint

  10. Du, K., Meng, Q.: A revisit to \(\mathrm{W}_{n}^{2}\)-theory of super-parabolic backward stochastic partial differential equations in ℝd. Stoch. Process. Appl. 120, 1996–2015 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Du, K., Tang, S.: Strong solution of backward stochastic partial differential equations in C2 domains. Probab. Theory Relat. Fields (2011). doi:10.1007/S00440-011-0369-0

    Google Scholar 

  12. Du, K., Tang, S., Zhang, Q.: Wm,p-solution (p≥2) of linear degenerate backward stochastic partial differential equations in the whole space (2011). arXiv:1105.1428v1

  13. Englezos, N., Karatzas, I.: Utility maximization with habit formation: dynamic programming and stochastic PDEs. SIAM J. Control Optim. 48, 481–520 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hu, Y., Peng, S.: Adapted solution of a backward semilinear stochastic evolution equations. Stoch. Anal. Appl. 9, 445–459 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hu, Y., Ma, J., Yong, J.: On semi-linear degenerate backward stochastic partial differential equations. Probab. Theory Relat. Fields 123, 381–411 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Johnson, W., Lindenstrauss, J. (eds.): Handbook of the Geometry of Banach Spaces, vol. 1. North-Holland, Amsterdam (2001)

    MATH  Google Scholar 

  17. Karoui, N.E., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558–602 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Krylov, N.V.: A Generalization of the Littlewood-Paley inequality with applications to parabolic equations. Ulam Q. 2, 16–26 (1994)

    MATH  Google Scholar 

  20. Krylov, N.V.: On L p -theory of stochastic partial differential equations. SIAM J. Math. Anal. 27, 313–340 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Krylov, N.V.: An analytic approach to SPDEs. In: Stochastic Partial Differential Equations: Six Perspectives. Mathematic Surveys and Monographs, vol. 64, pp. 185–242. AMS, Providence (1999)

    Google Scholar 

  22. Krylov, N.V.: On the Itô-Wentzell formula for distribution-valued processes and related topics. Probab. Theory Relat. Fields 150, 295–319 (2010)

    Article  Google Scholar 

  23. Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. J. Sov. Math. 16, 1233–1277 (1981)

    Article  MATH  Google Scholar 

  24. Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. AMS, Providence (1968)

    Google Scholar 

  25. Mikulevicius, R., Rozovskii, B.: A note on Krylov’s L p -theory for systems of SPDEs. Electron. J. Probab. 6, 1–35 (2001)

    Article  MathSciNet  Google Scholar 

  26. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Peng, S.: Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 30, 284–304 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Qiu, J., Tang, S.: On backward doubly stochastic differential evolutionary system (2010). Preprint

  29. Qiu, J., Tang, S.: Backward stochastic partial differential equations with degenerate, unbounded and irregular coefficients (2011). Preprint

  30. Qiu, J., Tang, S.: Maximum principles for backward stochastic partial differential equations (2011). Arxiv preprint arXiv:1103.1038

  31. Qiu, J., Tang, S., You, Y.: 2D backward stochastic Navier-Stokes equations with nonlinear forcing. Stoch. Process. Appl. (2011). doi:10.1016/j.spa.2011.08.010

    Google Scholar 

  32. Tang, S.: The maximum principle for partially observed optimal control of stochastic differential equations. SIAM J. Control Optim. 36, 1596–1617 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang, S.: Semi-linear systems of backward stochastic partial differential equations in ℝn. Chin. Ann. Math. 26B, 437–456 (2005)

    Article  Google Scholar 

  34. Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  35. Triebel, H.: Theory of Function Spaces II. Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992)

    Book  MATH  Google Scholar 

  36. Zhou, X.: A duality analysis on stochastic partial differential equations. J. Funct. Anal. 103, 275–293 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhou, X.: On the necessary conditions of optimal controls for stochastic partial differential equations. SIAM J. Control Optim. 31, 1462–1478 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Jinniao Qiu.

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Communicating Editor: Alain Bensoussan.

Supported by NSFC Grant #10325101, by Basic Research Program of China (973 Program) Grant # 2007CB814904, by the Science Foundation of the Ministry of Education of China Grant #200900071110001, and by WCU (World Class University) Program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (R31-2009-000-20007).

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Du, K., Qiu, J. & Tang, S. L p Theory for Super-Parabolic Backward Stochastic Partial Differential Equations in the Whole Space. Appl Math Optim 65, 175–219 (2012). https://doi.org/10.1007/s00245-011-9154-9

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