Forward-backward stochastic differential equations associated with systems of quasilinear parabolic equations and comparison theorems

We develop a probabilistic approach to construction of a viscosity solution of the Cauchy problem for a system of quasilinear parabolic equations with respect to a vector function $u(t,x)\in R^{d_1}$, $x\in R^d$. Our approach is based on a possibility to reduce the original quasilinear parabolic system to a quasilinear parabolic equation in an alternative phase space and derive forward-backward stochastic differential equations associated with it. This reduction shows the way to prove some comparison theorems for BSDEs and as a result to construct a probabilistic representation of a viscosity solution of the original Cauchy problem.