Forward and backward adapted quantum stochastic calculus and double product integrals

We show that iterated stochastic integrals can be described equivalently either by the conventional forward adapted, or by backward adapted quantum stochastic calculus. By using this equivalence, we establish two properties of triangular (causal) and rectangular double quantum stochastic product integrals, namely a necessary and sufficient condition for their unitarity, and the coboundary relation between the former and the latter.