associativity of stochastic integration

The chain ruleMathworldPlanetmath for expressing the derivativeMathworldPlanetmathPlanetmath of a variable z with respect to x in terms of a third variable y is


Equivalently, if dy=αdx and dz=βdy then dz=βαdx. The following theorem shows that the stochastic integral satisfies a generalizationPlanetmathPlanetmath of this.


Let X be a semimartingale and α be an X-integrable process. Setting Y=α𝑑X then Y is a semimartingale. Furthermore, a predictable process β is Y-integrable if and only if βα is X-integrable, in which case

β𝑑Y=βα𝑑X. (1)

Note that expressed in alternative notation, (1) becomes


or, in differentialMathworldPlanetmath notional,


That is, stochastic integration is associative.

Title associativity of stochastic integration
Canonical name AssociativityOfStochasticIntegration
Date of creation 2013-03-22 18:41:06
Last modified on 2013-03-22 18:41:06
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