Stopping Times of Boundaries: Relaxation and Continuity

H. Mete Soner, Valentin Tissot-Daguette

Published 2023-05-16Version 1

We study boundaries and their hitting times in the context of optimal stopping and Bermudan-style options. We prove the continuity of the map linking the boundary to the value of its associated stopping policy. While the supremum norm is used to compare continuous boundaries, a weaker topology induced by the so-called relaxed $L^{\infty}$ distance is used in the semicontinuous case. Relaxed stopping rules and their properties are also discussed. Incidentally, we provide a convergence analysis for neural stopping boundaries [Reppen, Soner, and Tissot-Daguette, Neural Optimal Stopping Boundary, 2022] entailing the universal approximation capability of neural networks and the notion of inf/sup convolution.