completeness under ucp convergence

Let (Ω,,(t)t+,) be a filtered probability space. Then, under the ucp topology, various classes of stochastic processesMathworldPlanetmath form completePlanetmathPlanetmathPlanetmathPlanetmath ( vector spaces.

For a σ-algebra on +×Ω, we can look at the processes which are -measurable when regarded as a map from +×Ω to . As the ucp topology is only defined for jointly measurable processes, we restrict attention to sub-σ-algebras of the productPlanetmathPlanetmathPlanetmath (+).

Theorem 1.

Let M be a sub-σ-algebra of B(R+)F. Then, the set of M-measurable processes is complete under ucp convergence.

That is, if Xn is a sequence of -measurable processes such that Xn-Xmucp0 as m,n then XnucpX for some -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 R+R which is closed ( under uniform convergenceMathworldPlanetmath on compactsPlanetmathPlanetmath (the compact-open topologyMathworldPlanetmath). Then, the set of jointly measurable processes whose sample paths are almost surely in S is complete under ucp convergence.

So, for example, the continuousMathworldPlanetmathPlanetmath, 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 (, the following useful result is obtained.


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