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 ${ℝ}^n$, 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