Noncommutative stochastic integration through decoupling

We present an abstract approach to noncommutative stochastic integration in the context of a finite von Neumann algebra equipped with a normal, faithful, tracial state, with respect to processes with tensor or freely independent increments satisfying a stationarity condition, using a decoupling technique. We obtain necessary and sufficient conditions for stochastic integrability of L p -processes with respect to such integrators. We apply the theory to stochastic integration with respect to Boson and free Brownian motion.