Gaussian density estimates for solutions of fully coupled forward-backward SDEs

TL;DR

This paper derives Gaussian density bounds for solutions of fully coupled one-dimensional forward-backward SDEs using PDE techniques, simplifying previous approaches by avoiding direct FBSDE manipulations.

Contribution

It introduces a PDE-based method to estimate densities of FBSDE solutions, providing a simpler alternative to existing techniques.

Findings

Established upper and lower Gaussian density estimates for FBSDE components.

Demonstrated the effectiveness of PDE methods over traditional FBSDE manipulations.

Simplified the analysis process for density estimation in coupled FBSDEs.

Abstract

We obtain upper and lower Gaussian density estimates for the law of each component of the solution to a one-dimensional fully coupled forward-backward SDE (FBSDE). Our approach relies on the link between FBSDEs and quasilinear parabolic PDEs, and is fully based on the use of classical results on PDEs rather than on manipulation of FBSDEs, compared to other papers on this topic. This essentially simplifies the analysis.