On Skorohod spaces as universal sample path spaces

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.