  In this paper, we formulate a general time-inconsistent stochastic
linear--quadratic (LQ) control problem. The time-inconsistency arises from the
presence of a quadratic term of the expected state as well as a state-dependent
term in the objective functional. We define an equilibrium, instead of optimal,
solution within the class of open-loop controls, and derive a sufficient
condition for equilibrium controls via a flow of forward--backward stochastic
differential equations. When the state is one dimensional and the coefficients
in the problem are all deterministic, we find an explicit equilibrium control.
As an application, we then consider a mean-variance portfolio selection model
in a complete financial market where the risk-free rate is a deterministic
function of time but all the other market parameters are possibly stochastic
processes. Applying the general sufficient condition, we obtain explicit
equilibrium strategies when the risk premium is both deterministic and
stochastic.
