No-arbitrage under a class of honest times (Q1691448)

For a market model to be financially and mathematically viable, it has to possess the NUPBR (no unbounded profit with bounded risk) property. It is known that even if the initial model \((S,\mathbb{F},P)\), given by a \(d\)-dimensional quasi-left-continuous semimartingale (modeling discounted asset prices) on a given stochastic basis \((\Omega, \mathcal{G}, \mathbb{F}, P)\) where \(\mathcal{F}_\infty:=\bigvee_{t\geq0}\mathcal{F}_t\subseteq\mathcal{G}\) and \(\mathbb{F}=(\mathcal{F}_t)_{t\geq0}\) is a filtration satisfying the usual hypotheses, satisfies NUPBR, trying to add new information in the form of a random time \(\tau\) can make the extended model \((S,\mathbb{F},\tau)\) lose this important property. Denoting the appropriately enlarged filtration by \(\mathbb{G}\) so that \(\tau\) is a \(\mathbb{G}\)-stopping time, the authors complement the investigation of \textit{A. Aksamit} et al. [Finance Stoch. 21, No. 4, 1103--1139 (2017; Zbl 1391.91165)], which focused on the process stopped at \(\tau\), namely \((S^\tau, \mathbb{G}\)), by looking into what happens after this random time, namely whether \((S-S^\tau, \mathbb{G})\) can still possess the NUPBR property. Since the process \(S-S^\tau\) might even lose the semimartingale property, one has to restrict attention to honest stopping times, but even this does not guarantee that the NUPBR property is preserved. By defining a new class of honest stopping times, which contains all \(\mathbb{F}\)-stopping times, as well as many examples of non-\(\mathbb{F}\)-stopping times, the authors show that for \(\tau\) from this class there are models for which NUPBR is preserved after \(\tau\). Moreover, the authors try to describe as explicitly as possible the part \(\mathcal{T}(S)\) of \(S\) which is truly relevant for the preservation of NUPBR, i.e., characterize the equivalence \[ (\mathcal{T}(S), \mathbb{F})\;\text{satisfies NUPBR if and only if}\;(S-S^\tau,\mathbb{G})\;\text{does}. \]