In a further generalisation of the classical Ito formula, we show that if B = (At + A[dagger]t [ratio ] t ∈ [0, 1]) is the Brownian quantum martingale and J = (Jt [ratio ] t ∈ [0, 1]) is a bounded quantum semimartingale, then M = (Bt + Jt [ratio ] t ∈ [0, 1]) satisfies the functional quantum Ito formula [10, Section 6]

formula here

The proof, which relies on the classical theory of unbounded self-adjoint operators, may be adapted to the case where B is replaced by a classical Brownian martingale.