= Quadratic variation
{title2=$[X]_t$}
{wiki}
For a <continuous semimartingale> $X$, its quadratic variation is the limit in <probability>
$$
[X]_t=\lim_{|\pi|\to0}\sum_{[u,v]\in\pi}(X_v-X_u)^2
$$
along deterministic <partitions of an interval> $[0,t]$ with mesh tending to zero. The process can be chosen continuous and nondecreasing. A continuous <finite-variation process> has zero quadratic variation, while standard <Brownian motion> satisfies $[W]_t=t$. Thus a <Brownian motion> path has fluctuation invisible to ordinary first-order change of variables.
Continuity is essential to the finite-variation assertion. A <càdlàg> <finite-variation process> $A$ has $[A]_t=\sum_{0<s\leq t}(\Delta A_s)^2$. For a <Poisson process> $N$ with unit jumps, $[N]_t=N_t$, although its paths have <finite variation>. The <Itô formula> uses these squared increments and jump corrections when changing variables.
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