Markovization of Randomized Stopping Times
Abstract
We study randomized stopping times for a Markov process, described by a progressively measurable intensity αt(ω). We prove that every admissible intensity has a Markovian representative λ(t,Xt), explicitly obtained from the observed measure and the surviving occupation measure. This representative preserves both the family of surviving mass measures and the joint distribution of the stopping time and the state at stopping. Whenever the relative entropy with respect to a reference intensity r(t,x) is finite, we prove an exact decomposition showing that Markovization does not increase the entropy. As a consequence, variational problems in which a randomized stopping time enters only through its observed measure and an entropy penalty can be reduced to optimization over Markovian intensities λ(t,x). For every observed measure admitting a finite-entropy representative, there is a unique minimum-entropy randomized stopping time, and it is Markovian. We discuss the connection with optimal Skorokhod embedding and prove a finite-constraint realization result for Brownian stopping. We also approximate arbitrary observed measures by those generated by bounded Markovian intensities.
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BibTeX
@misc{sokolov2026markovization,
title = {{Markovization of Randomized Stopping Times}},
author = {Kirill Sokolov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24652},
eprint = {2609.24652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24652},
}
RIS
TY - UNPB
TI - Markovization of Randomized Stopping Times
AU - Kirill Sokolov
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24652
UR - https://arxiv.org/abs/2609.24652
AB - We study randomized stopping times for a Markov process, described by a progressively measurable intensity α_t(ω). We prove that every admissible intensity has a Markovian representative λ(t,X_t), explicitly obtained from the observed measure and the surviving occupation measure. This representative preserves both the family of surviving mass measures and the joint distribution of the stopping time and the state at stopping. Whenever the relative entropy with respect to a reference intensity r(t,x) is finite, we prove an exact decomposition showing that Markovization does not increase the entropy. As a consequence, variational problems in which a randomized stopping time enters only through its observed measure and an entropy penalty can be reduced to optimization over Markovian intensities λ(t,x). For every observed measure admitting a finite-entropy representative, there is a unique minimum-entropy randomized stopping time, and it is Markovian. We discuss the connection with optimal Skorokhod embedding and prove a finite-constraint realization result for Brownian stopping. We also approximate arbitrary observed measures by those generated by bounded Markovian intensities.
ER -