On Skorohod spaces as universal sample path spaces

TL;DR

This paper demonstrates that Skorohod spaces serve as universal sample path spaces for RCLL processes, enabling factorization through standard Borel spaces and simplifying stochastic control problems.

Contribution

It extends the factorization theorem for RCLL processes, showing Skorohod spaces as universal sample path spaces with inheritable properties.

Findings

RCLL processes can be factored through Skorohod spaces.

Trajectories inherit properties like continuity and bounded variation.

Simplifies conditional expectations in stochastic control.

Abstract

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.