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For two zero-starting continuous martingales square-integrable at time $t$, the Itô isometry gives $\mathbb{E}(M_t\pm N_t{)}^2=\mathbb{E}[M\pm N{]}_t$. The polarization identity yields the displayed identity. Integrability of the cross-bracket follows from the Kunita-Watanabe inequality and the Cauchy-Schwarz inequality.

The polarization identity for quadratic covariation, together with $\mathbb{E}[M\pm N{]}_s=\mathbb{E}(M_s\pm N_s{)}^2$, yieldsThe bracket identity for each zero-starting square-integrable martingale is the Itô isometry for integration of the constant integrand against that martingale. Polarizing both identities gives the displayed covariance identity for continuous square-integrable martingales. The bracket is integrable: its total variation is at most $\sqrt{[M{]}_s[N{]}_s}$ by the Kunita-Watanabe inequality, whose expectation is finite by the Cauchy-Schwarz inequality. Therefore condition (iii) is equivalent to (ii).

For every real $\lambda$, the quadratic variation of $\lambda (M-M_s)+(N-N_s)$ on $[s,t]$ is nonnegative:Its discriminant is therefore nonpositive. This gives the pathwise Kunita-Watanabe inequality

For a partition $\pi$ of $[0,t]$, apply the first inequality to each increment of the covariation and then the Cauchy-Schwarz inequality:Taking the supremum over partitions proves the Kunita-Watanabe inequality

The Kunita-Watanabe inequality applied to the continuous local martingales $M,N$ gives directlyEquivalently, with the clock and densities from part (i), positivity of the covariance matrix gives $|c|\le \sqrt{ab}$, and the Cauchy-Schwarz inequality gives