  This paper is concerned with a linear-quadratic (LQ, for short) optimal
control problem for backward stochastic differential equations (BSDEs, for
short), where the coefficients of the backward control system and the weighting
matrices in the cost functional are allowed to be random. By a variational
method, the optimality system, which is a coupled linear forward-backward
stochastic differential equation (FBSDE, for short), is derived, and by a
Hilbert space method, the unique solvability of the optimality system is
obtained. In order to construct the optimal control, a new stochastic
Riccati-type equation is introduced. It is proved that an adapted solution
(possibly non-unique) to the Riccati equation exists and decouples the
optimality system. With this solution, the optimal control is obtained in an
explicit way.
