Indefinite stochastic linear-quadratic optimal control problems with random coefficients: Closed-loop representation of open-loop optimal controls

This paper is concerned with a stochastic linear-quadratic optimal control problem in finite time horizon, where the coefficients of system are allowed to be random, and weighting matrices cost functional random indefinite. It shown, Hilbert space approach, that for existence an open-loop control, convexity (with respect control) necessary; uniform convexity, which slightly stronger, turns out sufficient, also leads unique solvability associated Riccati equation. Further, it shown admits closed-loop representation. In addition, some sufficient conditions obtained functional, strictly more general than classical matrix-valued processes positive (semi-) definite.