generalized Ito formula

The generalized Itô formulaMathworldPlanetmathPlanetmath, or generalized Itô’s lemma, is an extensionPlanetmathPlanetmathPlanetmath of Itô’s lemma ( that applies also to discontinuousMathworldPlanetmath processes. For a cadlag process X, we write ΔXtXt-Xt- for its jump at time t.


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

df(Xs)=i=1nf,i(Xt-)dXti+12i,j=1nf,ij(Xt-)d[Xi,Xj]tc+(Δf(Xt)-i=1nf,i(Xt-)ΔXti). (1)

Here, [Xi,Xj]c represents the continuousMathworldPlanetmathPlanetmath part of the quadratic covariation,


which is a continuous finite variation process. The final term on the right hand side of (1) involving the jumps of X represents the differentialMathworldPlanetmath dZ of the process


This is indeed a well defined finite variation process, as the sum of the absolute valuesMathworldPlanetmathPlanetmathPlanetmath


is finite. Here, K is a finite random variableMathworldPlanetmath, and this bound follows from expanding f as a Taylor series to second orderPlanetmathPlanetmath.

The reason for using differential notation and writing the formula in terms of the continuous part of the quadratic covariation should be clear when it is considered that writing out the expression in full gives the following rather messy formula.


This formula may be understood as Itô’s formula ( for continuous processes together with an additional term to ensure that the jumps of the right hand side are equal to Δf(Xt). The need for this adjustment term comes from the fact that Itô’s formula for continuous processes is essentially a Taylor expansion to second order, which only applies when the increments δXt=Xt+δt-Xt vanish in the limit of small δt. This needs adjusting whenever the process jumps.

The first term on the right hand side of (1) is a stochastic integral and, hence, is a semimartingale. As the remaining terms are finite variation processes, the following consequence is obtained.


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

Title generalized Ito formula
Canonical name GeneralizedItoFormula
Date of creation 2013-03-22 18:41:47
Last modified on 2013-03-22 18:41:47
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 60G07
Classification msc 60H05
Classification msc 60H10
Synonym generalized Itô formula
Synonym generalized Itö formula
Synonym generalized Ito’s lemma
Synonym generalized Itô’s lemma
Synonym generalized Itö’s lemma
Related topic ItosLemma2
Related topic ItosFormula