Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs

This paper considers optimal (minimizing) control of stochastic linear quadratic regulators (LQRs). The assumption that the control weight costs must be positive definite, inherited from the deterministic case, has been taken for granted in the literature. It is, however, shown in this paper that some stochastic LQR problems with indefinite (in particular, negative) control weight costs may still be sensible and well-posed due to the deep nature of stochastic systems. New stochastic Riccati equations, which are backward stochastic differential equations involving complicated nonlinear terms, are presented and their solvability is proved to be sufficient for the well-posedness and the solutions of the optimal LQR problems. Existence and uniqueness of solutions to the Riccati equation for a special case are obtained. Finally, it is argued that, quite contrary to the deterministic systems, the stochastic maximum principle cannot fully characterize the optimality of the stochastic LQR problems.