Abstract. We show that for a quantum L-martingale (X(t)), p > 2, there exists a Doob-Meyer decomposition of the submartingale (|X(t)|). A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an L-martingale, p > 2, is constructed. Using this construction, the uniqueness of the Doob-Me...