  We study the properties of the free boundaries and the corresponding hitting
times in the context of optimal stopping in discrete time. We first prove the
continuity of the map from the boundaries to the expected value of the
corresponding stopping policy both in the supremum norm and also in a weaker,
novel topology induced by the relaxed $L^\infty$ distance that we introduce.
The latter is particularly useful when the optimal stopping boundary is only
proved to be semicontinuous. Secondly, we study the connection between the
hitting times, and their relaxations as widely employed in recent numerical
methods. All these results together with the universal approximation capability
of neural networks and the notion of inf/sup convolution are then used to
provide a convergence analysis for the algorithm introduced in [Reppen, Soner,
and Tissot-Daguette, Neural Optimal Stopping Boundary, 2022] for the numerical
resolution of the exercise regions arising in the analysis of Bermudan type
option.
