Well-posedness of backward stochastic differential equations with general filtration

This paper is addressed to the well-posedness of some linear and semilinear backward stochastic differential equations with general filtration, without using the Martingale Representation Theorem. The point of our approach is to introduce a new notion of solution, i.e., the transposition solution, which coincides with the usual strong solution when the filtration is natural but it is more flexible for the case of general filtration than the existing notion of solutions. A comparison theorem for transposition solutions and a Pontryagin-type stochastic maximum principle are also presented.