Quadratic variation

Remarks

A Wiener process $W$ has $[W]_t = t$. Check: the sum $Q_\pi$ has mean $\sum_k (t_k - t_{k-1}) = t$ and variance $2 \sum_k (t_k - t_{k-1})^2 \le 2t \norm{\pi}$, so $Q_\pi \to t$ in $L^2$ and hence in probability. A continuously differentiable function $f$ has $[f]_t = 0$, because $\sum_k (f(t_k) - f(t_{k-1}))^2 \le \max\abs{f'}^2 \, t \norm{\pi}$. For fractional Brownian motion with Hurst index $H < 1/2$, the sums over uniform partitions tend to infinity, so the limit does not exist. Every semimartingale has a quadratic variation, given by $[X]_t = X_t^2 - X_0^2 - 2 \int_0^t X_{s-} \, dX_s$. For a continuous local martingale $M$, $[M]$ is the unique continuous non-decreasing adapted process with $[M]_0 = 0$ such that $M^2 - [M]$ is a local martingale.