Quadratic variation | Glossary | Derivatives Compass

Quadratic variation is the value that the sum of the squared increments over the small subintervals of a partition approaches as the partition is made finer. The quadratic variation of Brownian motion equals the elapsed time, and that of a smooth function is 0.

For Brownian motion $W$ , as the partition of the interval $[0, T]$ is made finer, the sum of squared increments approaches $T$ with probability 1. This relation is written as the rule $(dW)^2 = dt$ for infinitesimal quantities.

Consider a hypothetical example in which $[0, 1]$ is divided into 100 equal parts. For a random walk with increments of $\pm 0.1$ , each squared increment is 0.01 and the sum of the 100 terms is 1. For the straight line $g(t) = t$ , each increment is 0.01, each square is 0.0001, and the sum is 0.01. As the partition is made finer, the sum for the line approaches 0, while the sum for Brownian motion stays at 1.

This property is the reason the extra term appears in Ito's lemma.

Related terms

Explained in detail