Non-homogeneous stochastic linear-quadratic optimal control problems with multi-dimensional state and regime switching

In this paper, we study non-homogeneous stochastic linear-quadratic (LQ) optimal control problems with multi-dimensional state and regime switching. We focus on the corresponding stochastic Riccati equation, which is the same as that one in homogeneous stochastic LQ optimal control problem, and the adjoint backward stochastic differential equation (BSDE), which arises from the non-homogeneous terms in the state equation and cost functional. Both stochastic Riccati equation and adjoint BSDE are solved by the contraction mapping method, and are used to represent the closed-loop optimal control and the optimal value of our problems.