We give a sufficient condition under which the stochastic optimal transportation problem is finite, which implies the existence of a semimartingale with given initial and terminal distributions. The idea of the proof is to show the finiteness of the supremum in the duality theorem for the stochastic optimal transportation problem. As a special case, it also gives a new approach for the construction of the h-path process with given initial and terminal distributions. We also consider a problem similar to the above for a class of optimal control problems for a family of solutions to Fokker-Planck equations.