quadratic variation

Suppose you want to measure 'how much' a wiggly path moves over a time interval. The naive measure adds up the lengths of all the little up-and-down moves — that is the total variation, the distance a tracing pen actually travels. For a Brownian path this is infinite: the path is so crinkly that the pen travels an unbounded distance even over one second. So we need a gentler ruler. Quadratic variation does the same accounting but adds up the SQUARES of the little moves instead of their lengths. Squaring tames the chaos, and the answer turns out to be beautifully simple and finite.

Here is the recipe. Chop the interval [0, t] into n equal small pieces. On each piece compute the change in B, square it, and add up all n squares: sum of [B(t_(k+1)) - B(t_k)]^2. As you make the pieces finer (n to infinity), this sum does NOT shrink to zero and does not blow up — it converges to exactly t. So the quadratic variation of Brownian motion over [0, t] equals t, written [B, B]_t = t. The reason is the variance bookkeeping: each squared increment over a piece of length t/n averages t/n, and there are n of them, so the sum averages n times (t/n) = t; and because the pieces are independent the random fluctuation around t washes out in the limit. A smooth curve, by contrast, has quadratic variation zero — its tiny changes are of order (t/n), whose squares are order (t/n)^2, summing to 0.

This single fact, [B, B]_t = t, is the engine of stochastic calculus. It is summarised by the heuristic (dB)^2 = dt: a squared Brownian increment behaves not like a negligible second-order term (as (dx)^2 does in ordinary calculus) but like a genuine first-order quantity, dt. That refusal to vanish is precisely why the ordinary chain rule breaks and why Ito's lemma carries an extra second-derivative term. Quadratic variation is also what distinguishes a real Brownian path from a smooth fake: it is the fingerprint of true roughness, and in finance it is the mathematical home of 'volatility'.