We present a new approach to noncommutative stochastic calculus that is, like
the classical theory, based primarily on the martingale property. Using this
approach, we introduce a general theory of stochastic integration and quadratic
(co)variation for a certain class of noncommutative processes -- analogous to
semimartingales -- that includes both the $q$-Brownian motions and classical $n
\times n$ matrix-valued Brownian motions. As applications, we obtain
Burkholder-Davis-Gundy inequalities (with $p \geq 2$) for continuous-time
noncommutative martingales and a noncommutative It\^{o} formula for "adapted
$C^2$ maps," including trace $\ast$-polynomial maps and operator functions
associated to the noncommutative $C^2$ scalar functions $\mathbb{R} \to
\mathbb{C}$ introduced by Nikitopoulos, as well as the more general
multivariate tracial noncommutative $C^2$ functions introduced by
Jekel-Li-Shlyakhtenko.
