Probabilistic Interpretation of the Cauchy Problem Solution for Systems of Nonlinear Parabolic Equations

Abstract

We reduce solution of the Cauchy problem for forward and backward systems of nonlinear parabolic equations to solution of certain stochastic problem formulated in terms of stochastic differential equations. Then we state existence and uniqueness theorems for solutions of the resulting stochastic systems and deduce existence and uniqueness of the Cauchy problem solutions for the original parabolic systems. As a by-product we derive stochastic representations of solutions to the Cauchy problem for some classes of nonlinear parabolic equation systems.