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 $Y\mathrm{=}\mathrm{\int }\alpha \mathit{}𝑑X$ then $Y$ is a semimartingale. Furthermore, a predictable process $\beta$ is $Y$-integrable if and only if $\beta \mathit{}\alpha$ 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
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