  In this paper, we propose a novel equilibrium solution notion for the
time-inconsistent stochastic linear-quadratic optimal control problem. This
notion is called the mixed equilibrium solution, which consists of two parts: a
pure-feedback-strategy part and an open-loop-control part. When the
pure-feedback-strategy part is zero or the open-loop-control part does not
depend on the initial state, the mixed equilibrium solution reduces to the
open-loop equilibrium control and the feedback equilibrium strategy,
respectively. Using a maximum-principle-like methodology with forward-backward
stochastic difference equations, a necessary and sufficient condition is
established to characterize the existence of a mixed equilibrium solution.
Then, by decoupling the forward-backward stochastic difference equations, three
sets of difference equations, which together portray the existence of a mixed
equilibrium solution, are obtained. Moreover, the case with a fixed time-state
initial pair and the case with all the initial pairs are separately
investigated. Furthermore, an example is constructed to show that the mixed
equilibrium solution exists for all the initial pairs, although neither the
open-loop equilibrium control nor the feedback equilibrium strategy exists for
some initial pairs.

Optimization and Control