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Probabilistic Counterparts of Nonlinear Parabolic Partial Differential Equation Systems

  • Yana I. Belopolskaya
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 90)

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.

Notes

Acknowledgements

Financial support of grant RFBR 12-01-00427-a and the Minobrnauki project 1.370.2011 is gratefully acknowledged

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.St. Petersburg State University for Architecture and Civil EngineeringSt. PetersburgRussia

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