# 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 $Y\mathrm{=}\mathrm{\int}\alpha \mathit{}\mathit{d}X$ 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

$$\int \beta \mathit{d}Y=\int \beta \alpha \mathit{d}X.$$ | (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 |