  This paper investigates the stochastic linear-quadratic (LQ, for short)
optimal control problems with non-Markovian regime switching in a finite time
horizon where the state equation is multi-dimensional. Similar to the classical
stochastic LQ problems, we establish the relationship between the stochastic LQ
optimal control problems with non-Markovian regime switching and the related
extended stochastic Riccati equations. To solve the extended stochastic Riccati
equations, we construct a monotone Piccard iterative sequence and present the
link bewteen this sequence and solutions of a family of forward-backward
stochastic differential equations. Relying on $L^p$ estimates for FBSDEs, we
show that the extended stochastic Riccati equation has a solution. This
partially addresses one question left in Hu et al. (Ann. Appl. Probab. 32(1):
426-460, 2022). Finally, the stochastic LQ optimal control problems with
non-Markovian regime switching is solved.
