Well-posedness of Stochastic Riccati Equations and Closed-Loop Solvability for Stochastic Linear Quadratic Optimal Control Problems

We study the closed-loop solvability of a stochastic linear quadratic optimal control problem for systems governed by stochastic evolution equations. This solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. At last, conditions ensuring the uniform convexity of the cost functional are discussed.