the quadratic covariation

When two martingales (or semimartingales) M and N evolve together, the quadratic covariation [M, N] measures how their increments move in tandem — the cross-variation analogue of [M] for a single process. It is what tells you whether two stochastic integrators are correlated, drives the cross terms in the multidimensional Ito formula and the Ito product rule, and reduces to the familiar instantaneous correlation for Brownian motions. It is the bilinear bracket that makes the space of martingales into something with an inner-product-like structure.

Definition by polarization or directly: [M, N]_t is the limit in probability, over partitions of [0,t] with mesh going to 0, of sum (M_{t_{i+1}} - M_{t_i})(N_{t_{i+1}} - N_{t_i}); equivalently [M, N] = (1/4)([M+N] - [M-N]). It is bilinear and symmetric, [M, M] = [M], and it satisfies the integration-by-parts / product rule M_t N_t = M_0 N_0 + integral M_{s-} dN_s + integral N_{s-} dM_s + [M, N]_t, which is exactly the Ito product rule — the bracket is the extra term beyond the two ordinary-looking integrals. Its jumps are Delta[M, N]_t = (Delta M_t)(Delta N_t), so for continuous processes [M, N] is continuous. The predictable counterpart <M, N> (the predictable covariation, from Doob-Meyer applied to the cross terms) is the predictable compensator of [M, N] and coincides with it when both processes are continuous.

Why it matters: in the multidimensional Ito formula d f(X) = sum partial_i f dX^i + (1/2) sum partial_{ij} f d[X^i, X^j], the second-order term is governed entirely by the matrix of covariations [X^i, X^j], so the covariation IS the local covariance structure of a diffusion. For independent Brownian motions [B^i, B^j]_t = delta_{ij} t (zero off the diagonal), the rigorous content of dB^i dB^j = delta_{ij} dt. The Kunita-Watanabe inequality bounds integrals against the covariation. The honest subtlety mirrors the single-process case: [M, N] is the optional/pathwise object and need not equal the predictable <M, N> when jumps are present; and two martingales can be uncorrelated in the sense [M, N] = 0 (then said to be orthogonal) without being independent — orthogonality is a covariation statement, not an independence statement.