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Probabilistic Counterparts of Nonlinear Parabolic Partial Differential Equation Systems

In: Modern Stochastics and Applications

Author

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  • Yana I. Belopolskaya

    (St. Petersburg State University for Architecture and Civil Engineering)

Abstract

We extend the results of the FBSDE theory in order to construct a probabilistic representation of a viscosity solution to the Cauchy problem for a system of quasilinear parabolic equations. We derive a BSDE associated with a class of quasilinear parabolic system and prove the existence and uniqueness of its solution. To be able to deal with systems including nondiagonal first order terms along with the underlying diffusion process, we consider its multiplicative operator functional. We essentially exploit as well the fact that the system under consideration can be reduced to a scalar equation in an enlarged phase space. This allows to obtain some comparison theorems and to prove that a solution to FBSDE gives rise to a viscosity solution of the original Cauchy problem for a system of quasilinear parabolic equations.

Suggested Citation

  • Yana I. Belopolskaya, 2014. "Probabilistic Counterparts of Nonlinear Parabolic Partial Differential Equation Systems," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 71-94, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-03512-3_5
    DOI: 10.1007/978-3-319-03512-3_5
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