| In recent years, stochastic linear-quadratic (LQ) control problem has been studied deeply and broadly. For deterministic LQ control problem, it is well-known that the control weighting matrix in the cost must be positive definite or at least positive semidefinite, otherwise the optimization problem will not be well-posed. However, it becomes quite different when stochastic LQ problem is investigated. This thesis is motivated by the discovery that the control weighting matrix can be indefinite if the diffusion term depends on the control in the stochastic case. It has not only theoretical significance but also interesting application especially in mathematical finance.; In the general form, the LQ problem with indefinite weighting matrices for a linear system subject to multiplicative noise can be solved via a "constrained" matrix Riccati equation.; For continuous-time indefinite LQ control problem, after introduce the generalized differential Riccati equation, we provide some necessary and/or sufficient conditions for its solvability. Furthermore, its asymptotic behavior is investigated along with connection to the corresponding generalized algebraic Riccati equation.; For discrete-time indefinite LCD control problem, by introducing the generalized discrete-time Riccati equation, we show that the well-posedness, the attainability of the LCD problem and the solvability of Riccati equation are all equivalent to each other. Moreover, in infinite time horizon, we introduce mean-square stabilizability and wellposedness. We also prove their conditions and present the optimal feedback control in terms of the maximal solution to discrete-time algebraic Riccati equation.; Finally, by applying existing semidefinite programming and linear matrix inequality tool, we carry out simulations to illustrate our theoretical results. |
