What conditions must we demand for the economy to be always on the saddle path?
To stick to the simpler deterministic case, it is perfect foresight and the Transvarsality condition (TVC)that guarantees that an optimizing agent will stay on the saddle-path, because all other paths lead to the violation of TVC (corner solutions violate the TVC).  In a deterministic environment and with perfect foresight the agent knows this and so he sticks to the saddle path. The chapter on the basic Ramsey model in Barrow and Sala-i-Martin's Growth  book has a good discussion on the matter.