Ito’s lemma

Itô’s lemma, also known as Itô’s formulaMathworldPlanetmathPlanetmath, is an extensionPlanetmathPlanetmath of the chain ruleMathworldPlanetmath ( to the stochastic integralMathworldPlanetmath, and is often regarded as one of the most important results of stochastic calculus. The case described here applies to arbitrary continuousMathworldPlanetmathPlanetmath semimartingales. For the application to Itô processes see Itô’s formula ( or see the generalized Itô formula ( for noncontinuous processes.

For a function f on a subset of n, we write f,i for the partial derivativeMathworldPlanetmath with respect to the i’th coordinate and f,ij for the second order derivatives.

Theorem (Itô).

Suppose that X=(X1,,Xn) is a continuous semimartingale taking values in an open subset U of Rn and f:UR is twice continuously differentiable. Then,

df(X)=i=1nf,i(X)dXi+12i,j=1nf,ij(X)d[Xi,Xj]. (1)

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


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 [Xi,Xj].

Equation (1) results from taking a Taylor expansion up to second orderPlanetmathPlanetmath which, setting δf(x)f(x+δx)-f(x), reads

δf(x)=i=1nf,i(x)δxi+12i,j=1nf,ij(x)δxiδxj+o(δx2). (2)

Taking the limit as δx goes to zero, all of the terms on the right hand side of (2), other than the first, go to zero with order ( O(δx2) and, therefore, can be neglected in the limit. This results in the standard chain rule. However, when δX=Xt+h-Xt 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 h0. This is a consequence of semimartingales, such as Brownian motionMathworldPlanetmath, being nowhere differentiableMathworldPlanetmath. 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