Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 38 / 2 / a / ii / Solution

The equal-time condition is the criterion proved equivalent to weak orthogonality in solution. Its equivalence with the quadratic-covariation condition follows from the covariance identity for continuous square-integrable martingales in solution; its equivalence with the forward-stopping condition follows from the optional sampling theorem in solution. These combine to give all four equivalences.