An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

TL;DR

This paper introduces an FBSDE-based method to solve the Skorokhod embedding problem for Gaussian processes with non-linear drift, extending existing theory to locally Lipschitz coefficients.

Contribution

It develops a novel approach using coupled FBSDEs and extends regularity theory for decoupling fields to non-globally Lipschitz coefficients.

Findings

Successfully solves the embedding problem for a class of Gaussian processes.

Extends FBSDE regularity theory to locally Lipschitz coefficients.

Provides new insights into the structure of solutions for non-linear drift processes.

Abstract

We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.