stochastic integration by parts

The stochastic integralMathworldPlanetmath satisfies a version of the classical integration by parts formulaMathworldPlanetmathPlanetmath, which is just the integral version of the product ruleMathworldPlanetmath. The only difference here is the existence of a quadratic covariation term.


Let X,Y be semimartingales. Then,

Xt⁒Yt=X0⁒Y0+∫0tXs-⁒𝑑Ys+∫0tYs-⁒𝑑Xs+[X,Y]t. (1)

Alternatively, in differentialMathworldPlanetmath notation, this reads


The existence of the quadratic covariation term [X,Y] in the integration by parts formula, and also in Itô’s lemma, is an important difference between standard calculus and stochastic calculus. To see the need for this term, consider the following. Choosing any h>0, write the increment of a process over a time step of size h as δ⁒Xt≑Xt+h-Xt. The increment of a productPlanetmathPlanetmath of processes satisfies the following simple identity,

δ⁒(X⁒Y)t=Xt⁒δ⁒Yt+Yt⁒δ⁒Xt+δ⁒Xt⁒δ⁒Yt. (2)

As we let h tend to zero, for differentiableMathworldPlanetmath processes the final term of (2) is of order ( O⁒(h2), so can be neglected in the limit. However, when X and Y are semimartingales, such as Brownian motionMathworldPlanetmath, the final term will be of order h, and needs to be retained even in the limit.

The proof of equation (1) is given by the proof of the existence of the quadratic variation of semimartingales ( and, in particular, is just a rearrangement of the formula given for the quadratic covariation of semimartingales. Whenever either of X or Y is a continuousMathworldPlanetmathPlanetmath finite variation process, the quadratic covariation term [X,Y] is zero, so (1) becomes the standard integration by parts formula. More generally, for noncontinuous processes we have the following.


Let X be a semimartingale and Y be an adapted finite variation process. Then,

Xt⁒Yt=X0⁒Y0+∫0tXs⁒𝑑Ys+∫0tYs-⁒𝑑Xs. (3)

As Y is a finite variation process, the first integral on the right hand side of (3) makes sense as a Lebesgue-Stieltjes integral. Equation (3) follows from the integration by parts formula by first substituting the following formula for the covariation whenever Y has finite variation into (1)


and then using the following identity

Title stochastic integration by parts
Canonical name StochasticIntegrationByParts
Date of creation 2013-03-22 18:41:35
Last modified on 2013-03-22 18:41:35
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Theorem
Classification msc 60H05
Classification msc 60G07
Classification msc 60H10