A property of two-parameter martingales with path-independent variation

Let M be a continuous two-parameter L4-martingale, vanishing on the axes, and f a 1-function. In Itô's formula for f(M2) a new martingale M is involved. This martingale can be interpreted formally as the stochastic integral [integral operator][not partial differential]1M[not partial differential]2M and it coincides with the martingale JM introduced by Cairoli and Walsh when M is strong. In this paper we prove that if M has path-independent variation, then M and M are orthogonal. Also. we give some counter-examples to the reciprocal implication.