Skip to main content
SpringerLink
Log in

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

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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 

Download references

Author information

Authors and Affiliations

  1. Department of Finance and Control Sciences, School of Mathematical Sciences, and Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai, 200433, China

    Kai Du, Jinniao Qiu & Shanjian Tang

  2. Graduate Department of Financial Engineering, Ajou University, San 5, Woncheon-dong, Yeongtong-gu, Suwon, 443-749, Korea

    Shanjian Tang

Authors
  1. Kai Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinniao Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shanjian Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinniao Qiu.

Additional information

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).

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-011-9154-9

Keywords

Search

Navigation