Characterization of Koenigs functions for category (ii) stopping models

Characterize the Koenigs functions associated with stopping models whose maximal parameter space is of the form $(0,\theta_{K_{\mathcal N}}]$ with $\theta_{K_{\mathcal N}}<1$, in order to determine the stopping models that ensure statistical stability of the generalized randomly stopped maximum/minimum transformations introduced by Valero and Ginebra (2025).

Background

The paper completely characterizes the stopping models with maximal parameter space $(0,1]$, including the identity stopping variable, through a bijective correspondence between such models and probability distributions supported on the positive integers. It also identifies a second class of connected stopping models whose maximal parameter space terminates at a value strictly below one; these models cannot include the identity as part of their connected component and are treated in Section 6.2.

The unresolved issue is to obtain a full characterization of the Koenigs functions governing this second class. Such a characterization would identify precisely which category (ii) stopping models support statistically stable generalized transformations combining randomly stopped maxima or minima with their functional inverses, and would provide a criterion for determining the maximal admissible parameter space directly from the Koenigs function.