Forward-backward stochastic differential equations with nonsmooth coefficients. (Q1877391)

The forward-backward SDE of the form \[ dX(t)=b(t,X(t),Y(t)) \,dt+\sigma (t,X(t),Y(t))\,dW(t), \quad t\in [0,T], \] \[ d Y(t)= h(t,X(t),Y(t), Z(t))\,dt +Z(t)\,dW(t), \quad t\in [0,T], \] \[ X(0)=x,\quad Y(T)=g(X(T)), \] is considered on a given probability space, where \(W\) is a standard \(n\)-dimensional Wiener process, the unknown process \((X(\cdot ),Y(\cdot ),Z(\cdot ))\) takes values in \(R^n\times R^m\times R^{m\times n}\) and the mappings \(b, \sigma , h, g \) are defined on respective spaces. Solvability of the equation is proved in multidimensional case under nondegeneracy conditions, for nonsmooth coefficients. If \(m=n=1\) the nondegeneracy of the forward diffusion is further relaxed, which leads to the existence of a viscosity solution to a degenerate nonsmooth quasilinear parabolic PDE.