= Solution
For every zero-starting <continuous local martingale> $M$, there is a unique, up to indistinguishability, continuous adapted increasing process $[M]$, with $[M]_0=0$, such that
$$
\boxed{M_t^2-[M]_t\text{ is a local martingale}.}
$$
This is its <quadratic variation>. It also satisfies, for deterministic partitions whose mesh tends to zero,
$$
[M]_t=\lim\sum_j\big(M_{t_{j+1}\wedge t}-M_{t_j\wedge t}\big)^2,
$$
with convergence uniformly on compact time intervals in probability. The theorem includes existence and finiteness on each compact interval; monotonicity is understood pathwise outside one null set.
Uniqueness follows because the difference of two candidate increasing continuous processes is both a <finite-variation process> and a <continuous local martingale>, hence is identically zero from its zero initial value. For a <continuous semimartingale> $X=X_0+M+A$, its <quadratic variation> is $[X]=[M]$, since continuous finite-variation terms have zero <quadratic variation>. Polarization defines the <quadratic covariation> by $[M,N]=([M+N]-[M-N])/4$.
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