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

In this paper, we explore non-homogeneous stochastic linear-quadratic (LQ) optimal control problems with multidimensional states and regime switching. We focus on the corresponding stochastic Riccati equation (SRE), which mirrors that of the 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. We solve both the SRE and adjoint BSDE using the contraction mapping method, which helps represent the closed-loop optimal control and the optimal value of our problems. In particular, we extend some results of Hu et al. [7] to the multidimensional case.