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  "body": "\n## Definition\n\nLet $(X_t)_{t \\ge 0}$ be a real [[stochastic process|0QH5ENKZM1]].\nFor a partition $\\pi = \\{0 = t_0 \u003c t_1 \u003c \\dots \u003c t_m = t\\}$ of $[0, t]$, let $\\norm{\\pi} = \\max_k (t_k - t_{k-1})$ be its mesh.\nThe process $X$ has *quadratic variation* $([X]_t)_{t \\ge 0}$ if for every $t \\ge 0$\n$$[X]_t = \\lim_{\\norm{\\pi} \\to 0} \\sum_{k=1}^{m} \\bigl(X_{t_k} - X_{t_{k-1}}\\bigr)^2,$$\nwhere the limit is taken in [[probability|0QA1HBD9Z4]] over partitions $\\pi$ of $[0, t]$.\n\n## Remarks\n\nA Wiener process $W$ has $[W]_t = t$.\nCheck: 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.\nA continuously differentiable function $f$ has $[f]_t = 0$, because $\\sum_k (f(t_k) - f(t_{k-1}))^2 \\le \\max\\abs{f'}^2 \\, t \\norm{\\pi}$.\nFor [[fractional Brownian motion|VC6R7XSKRQ]] with Hurst index $H \u003c 1/2$, the sums over uniform partitions tend to infinity, so the limit does not exist.\nEvery [[semimartingale|MZNFPJJMY9]] has a quadratic variation, given by $[X]_t = X_t^2 - X_0^2 - 2 \\int_0^t X_{s-} \\, dX_s$.\nFor a continuous [[local martingale|ZWJ9G89W3T]] $M$, $[M]$ is the unique continuous non-decreasing adapted process with $[M]_0 = 0$ such that $M^2 - [M]$ is a local martingale.\n",
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