$L^p$ estimations of fully coupled FBSDEs

TL;DR

This paper proves that under standard conditions, the unique $L^2$-solutions of fully coupled FBSDEs are also $L^p$-solutions for any $p>2$, extending previous estimations and solutions in stochastic control.

Contribution

It establishes a positive answer to whether $L^2$ solutions are also $L^p$ solutions for fully coupled FBSDEs, broadening the understanding of their solution spaces.

Findings

$L^p$ estimations are derived from $L^2$ estimations for FBSDEs.

Unique $L^p$-solutions exist under standard conditions.

Linear quadratic control problems with FBSDEs also admit unique $L^p$-solutions.

Abstract

In this study, for any given terminal time $T$, we establish an $L^p$ ($P>2$) estimations of fully coupled FBSDEs based on the $L^2$ estimations. Yong [24] proposed that a natural question is whether an adapted $L^2$-solution is an adapted $L^p$ solution for some $p>2$. In this study, we give a positive answer to this question. For any given terminal time $T$, based on an observation of the relation between $L^2$ and $L^p$ estimations of FBSDEs, we prove that a unique $L^2$-solution of fully coupled FBSDEs is an $L^p$-solution under standard conditions on the coefficients. Furthermore, we show that the fully coupled FBSDEs developed in the linear quadratic optimal control problem or investigated by the "decoupling random field" method admit a unique $L^p$-solution.