On Skorohod spaces as universal sample path spaces

The paper presents a factorization theorem for a certain class of stochastic processes. Skorohod spaces carry the rich structure of standard Borel spaces and appear to be suitable universal sample path spaces. We show that, if \(\xi\) is a RCLL stochastic process with values in a complete separable...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2004-12
Main Author: Delzeith, Oliver
Format: Article
Language:English
Subjects:
Existence theorems
Factorization
Metric space
Optimal control
Stochastic models
Stochastic processes
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The paper presents a factorization theorem for a certain class of stochastic processes. Skorohod spaces carry the rich structure of standard Borel spaces and appear to be suitable universal sample path spaces. We show that, if \(\xi\) is a RCLL stochastic process with values in a complete separable metric space \(E\), any other RCLL stochastic process \(X\) adapted to the filtration induced by \(\xi\) factors through the Skorohod space \(D_E[0,\infty)\). This can be understood as an extension of a stochastic process to a standard Borel space enjoying nice properties. Moreover, the trajectories of the factorized stochastic process defined on \(D_E[0,\infty)\) inherit the properties of being continuous, non-decreasing, and of bounded variation, resp., from those of \(X\). Considering situations which are invariant under the factorization procedure, the main theorem is a reduction tool to assume the underlying measurable space be a standard Borel space. In an example, we pick the existence theorem of regular conditional probabilities on standard Borel spaces to simplify a conditional expectation appearing in stochastic control problems.
ISSN:2331-8422