On Skorohod spaces as universal sample path spaces

2004/12/06 by Oliver Delzeith, Delzeith, Oliver
Mathematics · #60G07 (Primary) 60A10 (Secondary) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #math.OC #math.PR #msc:60A10 #msc:60G07

paper · pdf · doi:10.48550/arxiv.math/0412092

15 pages; submitted

arxiv created 2004/12/06 · arxiv updated 2009/12/01

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 ξ 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 ξ factors through the Skorohod space D_E[0,∞). 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,∞) 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.

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