 Quadratic variation

For a continuous semimartingale $X$, its quadratic variation is the limit in probabilityalong 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 $[A{]}_t={\sum }_{0<s\le t}(\Delta A_s{)}^2$. 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.