A Functional Approach to FBSDEs and Its Application in Optimal Portfolios

TL;DR

This paper introduces a functional differential equation approach to solving FBSDEs, enabling a unified solution process and application to optimal portfolio problems in incomplete markets.

Contribution

It extends the functional differential equations framework to solve FBSDEs in one direction and applies it to construct solutions for quadratic growth BSDE systems and an optimal portfolio problem.

Findings

Unified solution method for FBSDEs

Constructed solutions for quadratic BSDE systems

Solved an optimal portfolio problem in incomplete markets

Abstract

In Liang et al (2009), the current authors demonstrated that BSDEs can be reformulated as functional differential equations, and as an application, they solved BSDEs on general filtered probability spaces. In this paper the authors continue the study of functional differential equations and demonstrate how such approach can be used to solve FBSDEs. By this approach the equations can be solved in one direction altogether rather than in a forward and backward way. The solutions of FBSDEs are then employed to construct the weak solutions to a class of BSDE systems (not necessarily scalar) with quadratic growth, by a nonlinear version of Girsanov's transformation. As the solving procedure is constructive, the authors not only obtain the existence and uniqueness theorem, but also really work out the solutions to such class of BSDE systems with quadratic growth. Finally an optimal portfolio problem in incomplete markets is solved based on the functional differential equation approach and the nonlinear Girsanov's transformation.