Anna Aksamit

On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged Filtration

Given a reference filtration \(\mathbb{F}\), we consider the cases where an enlarged filtration \(\mathbb{G}\) is constructed from \(\mathbb{F}\) in two different ways: progressively with a random time or initially with a random variable. In both situations, under suitable conditions, we present a \(\mathbb{G}\)-optional semimartingale decomposition for \(\mathbb{F}\)-local martingales. Our study is then applied to the question of how an arbitrage-free semimartingale model is affected when stopped at the random time in the case of progressive enlargement or when the random variable used for initial enlargement satisfies Jacod’s hypothesis. More precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition, also called non arbitrages of the first kind in the literature. We provide alternative proofs of some results from Aksamit et al. (Non-arbitrage up to random horizon for semimartingale models, short version, preprint, 2014 [arXiv:1310.1142]), incorporating a different methodology based on our optional semimartingale decomposition.

No-arbitrage under a class of honest times

This paper quantifies the interplay between the no-arbitrage notion of no unbounded profit with bounded risk (NUPBR) and additional progressive information generated by a random time. This study complements the one of Aksamit et al. (Finance Stoch. 21:1103–1139, 2017) in which the authors have studied similar topics for the model stopped at the random time, while here we deal with the question of what happens after the random time. Given that the existing literature proves that NUPBR is always violated after honest times that avoid stopping times in a continuous filtration, we propose here a new class of honest times for which NUPBR can be preserved for some models. For these honest times, we obtain two principal results. The first result characterizes the pairs of initial market and honest time for which the resulting model preserves NUPBR, while the second result characterizes honest times that do not affect NUPBR of any quasi-left-continuous model (i.e., in which the asset price process has no predictable jump times). Furthermore, we construct explicitly local martingale deflators for a large class of models.

Projections, Pseudo-Stopping Times and the Immersion Property

Given two filtrations \(\mathbb{F} \subset \mathbb{G}\), we study under which conditions the \(\mathbb{F}\)-optional projection and the \(\mathbb{F}\)-dual optional projection coincide for the class of \(\mathbb{G}\)-optional processes with integrable variation. It turns out that this property is equivalent to the immersion property for \(\mathbb{F}\) and \(\mathbb{G}\), that is every \(\mathbb{F}\)-local martingale is a \(\mathbb{G}\)-local martingale, which, equivalently, may be characterised using the class of \(\mathbb{F}\)-pseudo-stopping times. We also show that every \(\mathbb{G}\)-stopping time can be decomposed into the minimum of two barrier hitting times.

No-arbitrage up to random horizon for quasi-left-continuous models

This paper studies the impact, on no-arbitrage conditions, of stopping the price process at an arbitrary random time. As price processes, we consider the class of quasi-left-continuous semimartingales, i.e., semimartingales that do not jump at predictable stopping times. We focus on the condition of no unbounded profit with bounded risk (called NUPBR), also known in the literature as no arbitrage of the first kind. The first principal result describes all the pairs of quasi-left-continuous market models and random times for which the resulting stopped model fulfils NUPBR. Furthermore, for a subclass of quasi-left-continuous local martingales, we construct explicitly martingale deflators, i.e., strictly positive local martingales whose product with the price process stopped at a random time is a local martingale. The second principal result characterises the random times that preserve NUPBR under stopping for any quasi-left-continuous model. The analysis carried out in the paper is based on new stochastic developments in the theory of progressive enlargements of filtrations.