  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.
