Quadratic variation is the second-order path statistic that survives when partition mesh vanishes. Continuous finite-variation motion contributes none; continuous local-martingale noise contributes all of it; jumps contribute their squared sizes. Covariation records common rough motion and drives the product and chain rules.
Partition limits
Definition. For a partition \(\pi=\{0=t_0<\cdots<t_m=t\}\), put \[ Q_\pi(X)_t=\sum_{i=0}^{m-1} (X_{t_{i+1}}-X_{t_i})^2. \] A cadlag process has quadratic variation \([X]\) if, along every deterministic partition sequence with mesh tending to zero, \(Q_\pi(X)\to[X]\) uniformly on compact time intervals in probability.
Proposition (finite variation). If \(A\) is continuous and has finite variation on compact intervals, then \[ [A]_t=0. \] If \(A\) is cadlag of finite variation, then \[ [A]_t=\sum_{0<s\le t}(\Delta A_s)^2. \]
Proof. For continuous \(A\), \[ \sum_i(\Delta A_i)^2 \le \max_i|\Delta A_i| \sum_i|\Delta A_i|. \] Uniform continuity makes the maximum tend to zero, while the second factor is bounded by total variation. For cadlag \(A\), isolate the finitely many jumps larger than \(\varepsilon\); the continuous and small-jump remainder is controlled by the same inequality. Let \(\varepsilon\downarrow0\).
Brownian and local-martingale variation
Theorem. For Brownian motion, \[ [B]_t=t. \] More generally, every continuous local martingale \(M\) has a unique continuous increasing adapted process \([M]\), starting at zero, such that \[ M_t^2-[M]_t \] is a local martingale.
Proof. For Brownian motion and a deterministic partition, \[ \mathbb EQ_\pi(B)_t=t,\qquad \operatorname{Var}(Q_\pi(B)_t) =2\sum_i(t_{i+1}-t_i)^2 \le2t|\pi|\to0. \] Thus \(Q_\pi(B)_t\to t\) in \(L^2\), and a maximal argument gives ucp convergence as a process. For a continuous local martingale, first stop so it is square integrable. Apply the discrete Doob decomposition to squared increments on refining partitions; the increasing compensators converge in probability, and the martingale parts converge by \(L^2\) estimates. Localization patches the limits. Uniqueness follows because a continuous finite-variation local martingale must be constant.
Definition. For a locally square-integrable martingale \(M\), its predictable bracket \(\langle M\rangle\) is the unique predictable increasing process such that \[ M^2-\langle M\rangle \] is a local martingale. For continuous local martingales, \([M]\) is continuous and adapted, hence predictable, so \[ [M]=\langle M\rangle. \] With jumps, optional quadratic variation and predictable bracket generally differ.
Semimartingale quadratic variation
Theorem. If \(X=X_0+M+A\) is a continuous semimartingale, with \(M\) a continuous local martingale and \(A\) continuous finite variation, then \[ [X]=[M],\qquad [X,A]=0. \]
Proof. Expanding partition sums gives \[ Q_\pi(X)=Q_\pi(M)+2\sum_i\Delta M_i\Delta A_i+Q_\pi(A). \] The last term tends to zero. Cauchy-Schwarz gives \[ \left|\sum_i\Delta M_i\Delta A_i\right| \le Q_\pi(M)^{1/2}Q_\pi(A)^{1/2}\to0 \] in probability because \(Q_\pi(M)\) is tight and \(Q_\pi(A)\to0\).
Covariation and polarization
Definition. For semimartingales \(X,Y\), \[ [X,Y]_t = \frac14\{[X+Y]_t-[X-Y]_t\}. \] Equivalently, \[ [X,Y]_t =\operatorname*{ucp-lim}_{|\pi|\to0} \sum_i\Delta X_i\Delta Y_i. \]
Proposition. Covariation is symmetric and bilinear, and \[ \Delta[X,Y]_t=\Delta X_t\Delta Y_t. \] For independent Brownian coordinates, \[ [B^i,B^j]_t=\delta_{ij}t. \]
Proof. Polarization and elementary algebra give bilinearity. A single jump contributes the product of the two jump sizes. For independent Brownian coordinates, cross-increment sums have mean zero and variance \[ \sum_i(\Delta t_i)^2\to0, \] while the diagonal case is Brownian quadratic variation.
Integration by parts
Theorem (product formula). For cadlag semimartingales, \[ X_tY_t = X_0Y_0 +\int_0^tX_{s-}\,\mathrm{d}Y_s +\int_0^tY_{s-}\,\mathrm{d}X_s +[X,Y]_t. \]
Proof. On each partition interval, \[ \Delta(XY)=X_{\mathrm{left}}\Delta Y +Y_{\mathrm{left}}\Delta X+\Delta X\Delta Y. \] Sum the exact identity. The first two sums converge to the stochastic integrals and the last to covariation.
The mnemonic \(\mathrm{d}B_t\,\mathrm{d}B_t=\mathrm{d}t\), \(\mathrm{d}t\,\mathrm{d}B_t=0\), and \((\mathrm{d}t)^2=0\) is shorthand for these partition limits, not an algebra of ordinary infinitesimals.