Indefinite stochastic linear quadratic control and generalized differential Riccati equation

We consider a stochastic linear-quadratic (LQ) problem with possible indefinite cost weighting matrices for the state and the control. An outstanding open problem is to identify an appropriate Riccati-type equation whose solvability is equivalent to the solvability of this possibly indefinite LQ problem. We introduce a new type of differential Riccati equation, called the generalized (differential) Riccati equation, which in turn provides a complete solution to the indefinite LQ problem. Moreover, all the optimal feedback/open-loop controls can be identified via the solution to this Riccati equation