Indefinite Stochastic Riccati Equations

This paper is concerned with stochastic Riccati equations (SREs), which are a class of matrix-valued, nonlinear backward stochastic differential equations (BSDEs). The SREs under consideration are, in general, indefinite, in the sense that certain parameter matrices are indefinite. This kind of equations arises from the stochastic linear-quadratic (LQ) optimal control problem with random coefficients and indefinite state and control weighting costs, the latter having profound implications in both theory and applications. While the solvability of the SREs is the key to solving the indefinite stochastic LQ control, it remains, in general, an extremely difficult, open problem. This paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite SREs for a number of special, yet important, cases.