  A discrete-time stochastic LQ problem with multiplicative noises and state
transmission delay is studied in this paper, which does not require any
definiteness constraint on the cost weighting matrices. From some abstract
representations of the system and cost functional, the solvability of this LQ
problem is characterized by some conditions with operator form. Based on these,
necessary and sufficient conditions are derived for the case with a fixed
time-state initial pair and the general case with all the time-state initial
pairs. For both cases, a set of coupled discrete-time Riccati-like equations
can be derived to characterize the existence and the form of the delayed
optimal control. In particular, for the general case with all the initial
pairs, the existence of the delayed optimal control is equivalent to the
solvability of the Riccati-like equations with some algebraic constraints, and
both of them are also equivalent to the solvability of a set of coupled linear
matrix equality-inequalities. Note that both the constrained Riccati-like
equations and the linear matrix equality-inequalities are introduced for the
first time in the literature for the proposed LQ problem. Furthermore, the
convexity and the {uniform convexity} of the cost functional are fully
characterized via certain properties of the solution of the Riccati-like
equations.
