Itō's Lemma

Itō's Lemma is a theorem of stochastic calculus that holds that within a closed integral, dz² can be replaced by dt, where dz is a stochastic variable with order of magnitude equal to the square root of dt. The Lemma is sometimes erroneously stated as "dz² equals dt," which is not generally true. Integration, with continuity, invokes the Law of Large Numbers. In the absence of continuity the variance of dz² is proportional to Δt, length of measurement intervals taken over the range.

Itō's Lemma is significant in finance because it provides the basis according to which a delta hedge is assumed to be riskless, an assumption that is essential to the Black-Scholes Equation.