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

¹ School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
² School of Mathematics, Shandong University, Jinan 250100, PR China
³ School of Economics, Shandong University, Jinan 250100, PR China

^* Corresponding author: zhangdetao@sdu.edu.cn

Received: 12 February 2022
Accepted: 3 November 2022

Abstract

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