  We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there exists a
Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. 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 $L^p$-martingale, $p>2$, is constructed. Using this construction, the
uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous
with probability one' is proved, and explicit forms of this decomposition and
the quadratic variation process for such a martingale are obtained.
