随机停时的马尔可夫化
Markovization of Randomized Stopping Times
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中文总结 AI 辅助
本文研究随机停时的马尔可夫化，证明任意可容许强度存在保持关键分布且不增熵的马尔可夫代表，从而将变分问题简化为对马尔可夫强度的优化，并给出唯一最小熵停时及布朗停时的实现结果。
中文摘要 AI 辅助
我们研究马尔可夫过程的随机停时，其由一个逐步可测的强度 $\alpha_t(\omega)$ 描述。我们证明每个可容许的强度都有一个马尔可夫代表 $\lambda(t,X_t)$，该代表可从观测测度和存活占据测度显式获得。此代表同时保留存活质量测度族以及停时与停止时状态的联合分布。每当相对于参考强度 $r(t,x)$ 的相对熵有限时，我们证明一个精确分解，表明马尔可夫化不会增加熵。因此，随机停时仅通过其观测测度和熵惩罚进入的变分问题可以简化为对马尔可夫强度 $\lambda(t,x)$ 的优化。对于每个允许有限熵代表的观测测度，存在唯一的最小熵随机停时，且该停时是马尔可夫的。我们讨论与最优Skorokhod嵌入的联系，并证明布朗停时的有限约束实现结果。我们还用有界马尔可夫强度生成的测度来近似任意观测测度。
英文摘要
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.
发表机构
- Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）
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