  This paper investigates the stochastic linear quadratic (LQ, for short)
optimal control problem of Markov regime switching system. The representation
of the cost functional for the stochastic LQ optimal control problem of Markov
regime switching system is derived using the technique of It{\^o}'s formula.
For the stochastic LQ optimal control problem of Markov regime switching
system, we establish the equivalence between the open-loop (closed-loop)
solvability and the existence of an adapted solution to the corresponding
forward-backward stochastic differential equation with constraint (the
existence of a regular solution to the Riccati equation). Also, we analyze the
interrelationship between the strongly regular solvability of the Riccati
equation and the uniform convexity of the cost functional.
