Time-inconsistent stochastic linear-quadratic control problem with indefinite control weight costs

A time-inconsistent linear-quadratic optimal control problem for stochastic differential equations is studied. We introduce conditions where the control cost weighting matrix is possibly singular. Under such conditions, we obtain a family of closed-loop equilibrium strategies via multi-person differential games. This result extends Yong’s work(2017) in the case of stochastic differential equations, where a unique closed-loop equilibrium strategy can be derived under standard conditions(namely, the control cost weighting matrix is uniformly positive definite, and the other weighting coefficients are positive semidefinite).