# Linear quadratic problems for fully coupled forward-backward stochastic control systems

## TL;DR

This paper develops a new approach to solve fully coupled forward-backward stochastic linear quadratic control problems with indefinite costs, deriving a state feedback form of the optimal control through novel decoupling and differential equations.

## Contribution

Introduces a new decoupling technique and non-Riccati-type ODEs to obtain the optimal control for fully coupled FBLQ problems with indefinite costs.

## Key findings

- Established existence of solutions for the derived ODEs.
- Derived the state feedback form of the optimal control.
- Illustrated results with special case examples.

## Abstract

This paper is concerned with optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs. In order to obtain the state feedback representation of the optimal control, we propose a new decoupling technique and obtain one kind of non-Riccati-type ordinary differential equations (ODEs). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs under some assumptions and derive the state feedback form of the optimal control. For this FBLQ problem, the optimal control depends on the entire trajectory of the state process. Some sepcial cases are given to illustrate our results.