  This paper is concerned with a backward stochastic linear-quadratic (LQ, for
short) optimal control problem with deterministic coefficients. The weighting
matrices are allowed to be indefinite, and cross-product terms in the control
and state processes are present in the cost functional. Based on a Hilbert
space method, necessary and sufficient conditions are derived for the
solvability of the problem, and a general approach for constructing optimal
controls is developed. The crucial step in this construction is to establish
the solvability of a Riccati-type equation, which is accomplished under a
fairly weak condition by investigating the connection with forward stochastic
LQ optimal control problems.
