# associativity of stochastic integration

The chain rule for expressing the derivative of a variable $z$ with respect to $x$ in terms of a third variable $y$ is

 $\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}.$

Equivalently, if $dy=\alpha\,dx$ and $dz=\beta\,dy$ then $dz=\beta\alpha\,dx$. The following theorem shows that the stochastic integral satisfies a generalization of this.

###### Theorem.

Let $X$ be a semimartingale and $\alpha$ be an $X$-integrable process. Setting $Y=\int\alpha\,dX$ then $Y$ is a semimartingale. Furthermore, a predictable process $\beta$ is $Y$-integrable if and only if $\beta\alpha$ is $X$-integrable, in which case

 $\int\beta\,dY=\int\beta\alpha\,dX.$ (1)

Note that expressed in alternative notation, (1) becomes

 $\beta\cdot(\alpha\cdot X)=(\beta\alpha)\cdot X$

or, in differential notional,

 $\beta(\alpha\,dX)=(\beta\alpha)\,dX.$

That is, stochastic integration is associative.

Title associativity of stochastic integration AssociativityOfStochasticIntegration 2013-03-22 18:41:06 2013-03-22 18:41:06 gel (22282) gel (22282) 4 gel (22282) Theorem msc 60H05 msc 60G07 msc 60H10