Skip to main content
SpringerLink
Log in

Probabilistic approach to viscosity solutions of the Cauchy problems for systems if fully nonlinear parabolic equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

In this paper, we discuss a probabilistic approach to construction of a viscosity solution of the Cauchy problem for a system of nonlinear parabolic equations. Our approach is based on reduction of the original problem to a system of quasilinear parabolic equation in the first step and to a system of fully coupled forward-backward stochastic differential equations in the second step. Solution of the stochastic problem allows us to construct a probabilistic representation of a viscosity solution of the original problem and state conditions which ensure the existence and uniqueness of this solution.

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

Access this article

Log in via an institution

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. Ya. Belopolskaya and Yu. Dalecky, “Investigation of the Cauchy problem for systems of quasilinear equations via Markov processes,” Izv. VUZ, Matematika, 12, 6-17 (1978).

    Google Scholar 

  2. Ya. Belopolskaya and Yu. L. Dalecky, Stochastic Equations and Differential Geometry, Kluwer (1990).

  3. Ya. Belopolskaya and Yu. Dalecky, “Markov processes associated with nonlinear parabolic systems,” Dokl. Akad. Nauk SSSR, 250, 521-524 (1980).

    MathSciNet  Google Scholar 

  4. J. F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York (1978).

    MATH  Google Scholar 

  5. Ya. Belopolskaya, “Probability approach to solution of nonlinear parabolic equations,” Probl. Math. Anal., 13, 21-35 (1992).

    Google Scholar 

  6. E. Pardoux and S. Peng, “Adapted solution of a backward stochastic differential equation,” Syst. Contr. Lett., 14, 55-61 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Pardoux and S. Peng, Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, Lect. Notes CIS, 176, Springer-Verlag (1992).

  8. S. Peng “Probabilistic interpretation for systems of qualilinear parabolic partial differential equations,” Stoch. Stoch. Rep., 37, 61-74 (1991).

    MATH  Google Scholar 

  9. J. Ma and J. Yong, “Forward-backward stochastic differential equations and their applications,” Lect. Notes Math., 1702 (1999).

  10. H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications, Ser. Stoch. Modeling Appl. Probab., 61, Springer (2009).

  11. P. Cheredito, H. Soner, N. Tuozi, and N. Victoir, “Second order backward stochastic differential equations and fully nonlinear parabolic PDEs,” Comm. Pure Appl. Math., 60, 1081-1110 (2007).

    Article  MathSciNet  Google Scholar 

  12. M. Crandall, H. Ishii, and P. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Amer. Math. Soc., 27, 1-67 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  13. Ya. Belopolskaya, “Probabilistic approach to nonlinear parabolic equations in jet-bundles,” Global Stochastic Anal., 1, 3-40 (2011).

    Google Scholar 

  14. Y. Hu and S. Peng, “Solutions of forward-backward stochastic differential equations,” Probab. Theory Relat. Fields, 103, 273-283 (1995).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University for Architectural and civil Engineering, St. Petersburg, Russia

    Ya. I Belopolskaya

  2. Case Western Reserve University, Cleveland, USA

    W. A. Woyczynski

Authors
  1. Ya. I Belopolskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. A. Woyczynski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya. I Belopolskaya.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 396, 2011, pp. 31-66.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belopolskaya, Y.I., Woyczynski, W.A. Probabilistic approach to viscosity solutions of the Cauchy problems for systems if fully nonlinear parabolic equations. J Math Sci 188, 655–672 (2013). https://doi.org/10.1007/s10958-013-1155-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1155-6

Keywords

Search

Navigation