Itô calculus

Summary

Itô calculus is the calculus of integrals with respect to Brownian motion and, more generally, semimartingales. Its central object is the Itô integral $\int_0^t H_s \, dB_s$ of an adapted process $H$. Its chain rule is Itô's lemma, which has an extra second-order term coming from the quadratic variation of the path. It is the language of stochastic differential equations and of continuous-time models in finance and physics.