The method of decoupling fields generalized to higher spatial derivatives

TL;DR

This paper extends the method of decoupling fields to higher spatial derivatives in strongly coupled forward-backward stochastic differential equations, analyzing their backward dynamics and conditions for singularities.

Contribution

It generalizes the decoupling field method to higher derivatives and investigates their backward behavior and singularity conditions in stochastic differential equations.

Findings

Derived backward dynamics for higher derivatives.

Identified conditions for singularity formation.

Extended the theoretical framework for decoupling fields.

Abstract

This work studies the spatial derivatives of decoupling fields to strongly coupled forward-backward stochastic differential equations in a Brownian setting. We formally deduce the backward dynamics of the first and higher spatial derivatives. In addition, we study necessary conditions under which singularities in either one of them can occur while moving backwards in time.