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

In this paper, we discuss a probabilistic approach to the construction of a viscosity solution of the Cauchy problem for a system of nonlinear parabolic equations. Our approach is based on a 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. The solution of the stochastic problem allows us to construct a probabilistic representation of a viscosity solution of the original problem and state conditions to ensure the existence and uniqueness of this solution.