Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations

TL;DR

This paper explores two equivalent families of linear fully coupled FBSDEs, establishing conditions for well-posedness and demonstrating how transformations can reduce coupling while preserving solutions, with applications to LQ problems.

Contribution

It introduces a linear transformation approach to relate two families of FBSDEs, ensuring well-posedness and solution determination without losing generality.

Findings

Equivalent families of FBSDEs are identified.

Transformation reduces coupling without affecting solutions.

Application to an optimal Linear Quadratic problem.

Abstract

In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDE). Within these families, one could get the same well-posedness of FBSDEs with totally different structures. The first family of FBSDEs are proved to be equivalent with respect to the Unified Approach. Thus one could get the well-posedness of the whole family if one member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. By reason of the fully coupling structure between the forward and backward equations, it leads to a highly interdependence in solutions. We are able to lower the coupling of FBSDEs, by virtue of the idea of transformation, without losing the well-posedness. Moreover, owing to the non-degeneracy of the transformation matrix, the solution to original FBSDE is totally determined by solutions of FBSDE after transformation. In addition, an example of optimal Linear Quadratic (LQ) problem is presented to illustrate.