# completeness under ucp convergence

Let $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\mathbb{R}}_{+}},\mathbb{P})$ be a filtered probability space. Then, under the ucp topology, various classes of stochastic processes^{} form complete^{} (http://planetmath.org/Complete) vector spaces.

For a $\sigma $-algebra $\mathcal{M}$ on ${\mathbb{R}}_{+}\times \mathrm{\Omega}$, we can look at the processes which are $\mathcal{M}$-measurable when regarded as a map from ${\mathbb{R}}_{+}\times \mathrm{\Omega}$ to $\mathbb{R}$. As the ucp topology is only defined for jointly measurable processes, we restrict attention to sub-$\sigma $-algebras of the product^{} $\mathcal{B}({\mathbb{R}}_{+})\otimes \mathcal{F}$.

###### Theorem 1.

Let $\mathrm{M}$ be a sub-$\sigma $-algebra of $\mathrm{B}\mathit{}\mathrm{(}{\mathrm{R}}_{\mathrm{+}}\mathrm{)}\mathrm{\otimes}\mathrm{F}$. Then, the set of $\mathrm{M}$-measurable processes is complete under ucp convergence.

That is, if ${X}^{n}$ is a sequence of $\mathcal{M}$-measurable processes such that ${X}^{n}-{X}^{m}\stackrel{ucp}{\to}0$ as $m,n\to \mathrm{\infty}$ then ${X}^{n}\stackrel{ucp}{\to}\mathrm{X}$ for some $\mathcal{M}$-measurable process $X$.

In particular, the spaces of jointly measurable, progressive, optional and predictable processes are each complete under ucp convergence. We can also look at the properties of the sample paths of the processes.

###### Theorem 2.

Let $S$ be any set of functions ${\mathrm{R}}_{\mathrm{+}}\mathrm{\to}\mathrm{R}$ which is closed (http://planetmath.org/Closed) under uniform convergence^{} on compacts^{} (the compact-open topology^{}). Then, the set of jointly measurable processes whose sample paths are almost surely in $S$ is complete under ucp convergence.

So, for example, the continuous^{}, right-continuous, left-continuous and cadlag processes are each complete under the ucp topology. Furthermore, combining theorems 1 and 2, and using the fact that a cadlag process is adapted if and only if it is jointly measurable (see measurability of stochastic processes (http://planetmath.org/MeasurabilityOfStochasticProcesses)), the following useful result is obtained.

###### Corollary.

The space of cadlag adapted processes is complete under ucp convergence.

Title | completeness under ucp convergence |
---|---|

Canonical name | CompletenessUnderUcpConvergence |

Date of creation | 2013-03-22 18:40:33 |

Last modified on | 2013-03-22 18:40:33 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 60G07 |

Classification | msc 60G05 |