# dominated convergence for stochastic integration

The dominated convergence theorem for standard integration states that if a sequence of measurable functions^{} converge to a limit, and are dominated by an integrable function, then their integrals converge to the integral of the limit. That is, the limit commutes with integration. A similar result holds for stochastic integration with respect to a semimartingale $X$, except the integrals are random variables^{}, and the integrals converge in probability.

###### Theorem (Dominated convergence).

If ${\xi}^{n}$ are predictable processes converging pointwise to $\xi $, and $\mathrm{|}{\xi}^{n}\mathrm{|}\mathrm{\le}\alpha $ for every $n$ and some $X$-integrable process $\alpha $, then

$${\int}_{0}^{t}{\xi}^{n}\mathit{d}X\to {\int}_{0}^{t}\xi \mathit{d}X$$ | (1) |

in probability as $n\mathrm{\to}\mathrm{\infty}$. Furthermore, ucp convergence and semimartingale convergence hold.

Note that as $\xi $ and ${\xi}^{n}$ are bounded^{} by an $X$-integrable process, they are guaranteed to also be $X$-integrable. Convergence in probability for each $t$ was taken as part of the definition of the stochastic integral, but the dominated convergence theorem stated here says that the stronger ucp and semimartingale convergence also hold.

If $\alpha $ is a locally bounded (http://planetmath.org/LocalPropertiesOfProcesses) predictable process, then it is automatically $X$-integrable for any semimartingale $X$. It follows that if ${\xi}^{n}$ are predictable processes converging to $\xi $ and if ${sup}_{n}|{\xi}^{n}|$ is locally bounded then the limit (1) holds. This result is sometimes known as the *locally bounded convergence theorem*.

To prove this result, it is enough to show that semimartingale convergence holds, as semimartingale convergence implies ucp convergence. So, let $|{\alpha}^{n}|\le 1$ be a sequence of simple predictable processes and set ${Y}^{n}=\int {\xi}^{n}\mathit{d}X$, $Y=\int \xi \mathit{d}X$. Associativity of stochastic integration gives

$${\int}_{0}^{t}{\alpha}^{n}\mathit{d}{Y}^{n}-{\int}_{0}^{t}{\alpha}^{n}\mathit{d}Y={\int}_{0}^{t}{\alpha}^{n}({\xi}^{n}-\xi )\mathit{d}X$$ |

However, $|{\alpha}^{n}({\xi}^{n}-\xi )|\le 2\alpha $, which is $X$-integrable. So, this converges to zero in probability by the definition of the stochastic integral, and ${Y}^{n}\to Y$ in the semimartingale topology.

Title | dominated convergence for stochastic integration |
---|---|

Canonical name | DominatedConvergenceForStochasticIntegration |

Date of creation | 2013-03-22 18:41:03 |

Last modified on | 2013-03-22 18:41:03 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 8 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 60H10 |

Classification | msc 60G07 |

Classification | msc 60H05 |

Defines | locally bounded convergence theorem |