A score-based test, built on Doob's h-transform, gives necessary and sufficient conditions for almost sure controlled set invariance of Itô diffusions over finite and infinite horizons.
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Set Invariance with Probability One for Controlled Diffusion: Score-based Approach
A score-based test, built on Doob's h-transform, gives necessary and sufficient conditions for almost sure controlled set invariance of Itô diffusions over finite and infinite horizons.