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. In this paper 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.