Brownian paths and quadratic variation

A standard one-dimensional Brownian motion starts at zero, has continuous paths almost surely, and has Gaussian increments of mean zero and variance equal to elapsed time. Thus the typical increment over a short interval is of square-root time scale, rather than proportional to time. Independent increments are asserted relative to the chosen filtration, not merely relative to the displayed process values. This is the information condition that later makes left-endpoint stochastic sums have mean zero.

Brownian scaling says that $c^{-1/2}W_{ct}$ is again standard Brownian motion for positive $c$. It preserves distributions of complete paths, not equality of a particular rescaled path with the original one. A deterministic piecewise-linear drawing can approximate Brownian paths in a suitable probabilistic construction, but each individual drawing has finite variation. A property shared by every finite approximation need not survive its limit: total variation is a central example of this mismatch.

Fix a finite horizon and a deterministic partition. Sum the squared Brownian increments over its intervals. Independence gives mean equal to the horizon, and the fourth-moment identity gives variance twice the sum of squared interval lengths. This variance is bounded by twice the horizon times the mesh. Therefore, along any sequence of deterministic partitions whose mesh tends to zero, the sums converge in mean square, and hence in probability, to the horizon.

For a fixed horizon and its dyadic equal subdivisions, the variance bounds are summable. Chebyshev's inequality and the Borel–Cantelli lemma then strengthen the conclusion to almost-sure convergence along that sequence. This does not assert one common probability-one event on which every conceivable sequence of path-dependent partitions gives the same limit. Adaptively choosing intervals from a fully observed path requires separate hypotheses and can change the statement. Convergence mode and permitted partition class belong to the theorem.

For a continuous function of finite total variation, a squared-increment sum is bounded by its largest increment times its total variation. Uniform continuity drives the first factor to zero as the mesh shrinks. Its quadratic variation along such partitions is therefore zero. Brownian dyadic quadratic variation is positive on every fixed interval of positive length, so Brownian paths cannot have finite total variation there. This argument explains the obstruction without pretending to differentiate a typical Brownian path.

The process identity $[W]_t=t$ supplies the extra clock in Itô calculus. For a diffusion with a drift term and a Brownian coefficient, the drift contributes no continuous quadratic variation, while the squared Brownian coefficient controls it. That later statement uses stochastic integration, not a rule for multiplying infinitesimals as ordinary real numbers. The familiar shorthand that the square of a Brownian differential equals elapsed time summarizes a proved limiting calculation; it is not a replacement for specifying the limit.