In this talk, we investigate some properties on Z, the martingale integrant of the backward stochastic differential equations (BSDEs in short), which is the sec ond process of the solution.These properties include bounded property, the back ward stochastic viability property (BSVP in short) and the comparison theorem for Z.Then we apply the obtained theoretical results in mathematics finance and a stochastic optimal control problem under Non-Markovian frame.