  In this paper we prove a weak necessary and sufficient maximum principle for
Markovian regime switching stochastic optimal control problems. Instead of
insisting on the maximum condition of the Hamiltonian, we show that 0 belongs
to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's
normal cone of the control constraint set at the optimal control. Under a joint
concavity condition on the Hamiltonian and a convexity condition on the
terminal objective function, the necessary condition becomes sufficient. We
give four examples to demonstrate the weak stochastic maximum principle.
