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

TL;DR

This paper introduces a Newton-Kantorovitch iterative method for solving decoupled forward-backward stochastic differential equations with smooth, bounded coefficients, demonstrating linear convergence under suitable initial conditions.

Contribution

The paper develops a Newton-Kantorovitch approach tailored for decoupled FBSDEs and establishes convergence criteria based on solutions of linear backward stochastic differential equations.

Findings

Method converges linearly to the solution.

Suitable initial conditions are characterized by linear backward SDEs.

Applicable to FBSDEs with smooth, bounded coefficients.

Abstract

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.