Newton-Kantorovitch method for decoupled forward-backward stochastic differential equations

We present and prove a Newton-Kantorovitch method for solving decoupled forward-backward stochastic differential equations (FBSDEs) involving smooth coefficients with uniformly bounded derivatives. As Newton's method is required a suitable initial condition to converge, we show that such initial conditions are solutions of a linear backward stochastic differential equation. In addition, we show that converges linearly to the solution.