The cocycle identity holds under stopping

In recent work of the authors, it was shown how to use any finite quantum stop time to stop the CCR flow and its strongly continuous isometric cocycles (Q. J. Math. 65:1145-1164, 2014). The stopped cocycle was shown to satisfy a stopped form of the cocycle identity, valid for deterministic increments of the time used for stopping. Here, a generalisation of this identity is obtained, where both cocycle parameters are replaced with finite quantum stop times.