Probabilistic Counterparts of Nonlinear Parabolic Partial Differential Equation Systems

Abstract

AbstractWe 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.KeywordsParabolic Partial Differential Equation SystemsBackward Stochastic Differential Equations (BSDE)Solution ViscosityComparison TheoremScalar Cauchy ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.