# Ito’s lemma

Itô’s lemma, also known as *Itô’s formula ^{}*, is an extension

^{}of the chain rule

^{}(http://planetmath.org/ChainRuleSeveralVariables) to the stochastic integral

^{}, and is often regarded as one of the most important results of stochastic calculus. The case described here applies to arbitrary continuous

^{}semimartingales. For the application to Itô processes see Itô’s formula (http://planetmath.org/ItosFormula) or see the generalized Itô formula (http://planetmath.org/GeneralizedItoFormula) for noncontinuous processes.

For a function $f$ on a subset of ${\mathbb{R}}^{n}$, we write ${f}_{,i}$ for the partial derivative^{} with respect to the $i$’th coordinate and ${f}_{,ij}$ for the second order derivatives.

###### Theorem (Itô).

Suppose that $X\mathrm{=}\mathrm{(}{X}^{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{X}^{n}\mathrm{)}$ is a continuous semimartingale taking values in an open subset $U$ of ${\mathrm{R}}^{n}$ and $f\mathrm{:}U\mathrm{\to}\mathrm{R}$ is twice continuously differentiable. Then,

$$df(X)=\sum _{i=1}^{n}{f}_{,i}(X)d{X}^{i}+\frac{1}{2}\sum _{i,j=1}^{n}{f}_{,ij}(X)d[{X}^{i},{X}^{j}].$$ | (1) |

In particular, for a continuous real-valued semimartingale $X$, (1) becomes

$$df(X)={f}^{\prime}(X)dX+\frac{1}{2}{f}^{\prime \prime}(X)d[X],$$ |

which is a form of the “change of variables formula” for stochastic calculus. A major distinction between standard and stochastic calculus is that here we need to include the quadratic variation and covariation terms $[X]$ and $[{X}^{i},{X}^{j}]$.

Equation (1) results from taking a Taylor expansion up to second order^{} which, setting $\delta f(x)\equiv f(x+\delta x)-f(x)$, reads

$$\delta f(x)=\sum _{i=1}^{n}{f}_{,i}(x)\delta {x}^{i}+\frac{1}{2}\sum _{i,j=1}^{n}{f}_{,ij}(x)\delta {x}^{i}\delta {x}^{j}+o(\delta {x}^{2}).$$ | (2) |

Taking the limit as $\delta x$ goes to zero, all of the terms on the right hand side of (2), other than the first, go to zero with order (http://planetmath.org/LandauNotation) $O(\delta {x}^{2})$ and, therefore, can be neglected in the limit. This results in the standard chain rule. However, when $\delta X={X}_{t+h}-{X}_{t}$ for a semimartingale $X$ then the second order terms in (2) only go to zero at rate $O(h)$ and, therefore, must be retained even in the limit as $h\to 0$. This is a consequence of semimartingales, such as Brownian motion^{}, being nowhere differentiable^{}.
In fact, if $X$ is a finite variation process, then it can be shown that the quadratic covariation terms are zero, and the standard chain rule results.

A consequence of Itô’s lemma is that if $X$ is a continuous semimartingale and $f$ is twice continuously differentiable, then $f(X)$ will be a semimartingale. However, the generalized Itô formula shows that it is not necessary to restrict this statement to continuous processes.

Title | Ito’s lemma |

Canonical name | ItosLemma |

Date of creation | 2013-03-22 18:41:44 |

Last modified on | 2013-03-22 18:41:44 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 60H10 |

Classification | msc 60G07 |

Classification | msc 60H05 |

Synonym | Itô’s lemma |

Synonym | Itö’s lemma |

Synonym | Ito’s formula |

Synonym | Itô’s formula |

Synonym | Itö’s formula |

Related topic | ItosFormula |

Related topic | GeneralizedItoFormula |