---
tag: JA95NH206E
kind: concept
name: Itô calculus
msc: [60H05, 60H10]
sources:
  - url: https://en.wikipedia.org/wiki/It%C3%B4_calculus
same_as:
  wikidata: Q947053
  wikipedia: Itô calculus
---

## Summary

Itô calculus is the calculus of integrals with respect to [[Brownian motion|22735FGJV6]] and, more generally, [[semimartingales|4EJB6E2AZK]].
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|2J1RVQB0CM]], which has an extra second-order term coming from the [[quadratic variation|X4DHMZD5JA]] of the path.
It is the language of stochastic differential equations and of continuous-time models in finance and physics.

## Explanations

### L1 human

A Brownian path is too rough to have a derivative, so the usual integral $\int H \, dB$ cannot be defined path by path.
Itô's idea is to approximate by sums $\sum H_{t_i} (B_{t_{i+1}} - B_{t_i})$ that always use the value of $H$ at the left end of each interval.
Such a sum is a fair bet: the stake is fixed before the random increment is seen.
The limit of these sums is the Itô integral.
For example, $\int_0^t B_s \, dB_s = \tfrac12 (B_t^2 - t)$, and the term $-t/2$ shows that the rules of ordinary calculus change.