AKSAMIT Anna Natalia

The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration F. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all F-stopping times. Then, for a given random time τ , we introduce F τ , the smallest right-continuous filtration containing F and making τ a stopping time, and we show that, for a thin time τ , each F-martingale is an F τ-semimartingale, i.e., the hypothesis (H) for (F, F τ) holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglb\"ock et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.

This paper completes the studies undertaken in [3, 4] and [8], where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called NUPBR hereafter) for quasi-left-continuous models and discrete-time market models respectively. Herein, we focus on the NUPBR for semimartingales models that live on thin predictable sets only and when the extra information about the random time is added progressively over time. For this setting, we explain how far the NUPBR property is affected when one stops the model by an arbitrary random time or when one incorporates fully an honest time into the model. Furthermore, we show how to construct explicitly local martingale deflator under the bigger filtration from those of the smaller filtration. As consequence, by combining the current results on the thin case and those of [3, 4], we elaborate universal results for general semimartingale models.

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping with the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature —up to our knowledge— proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore , we construct explicitly " the-after-τ " local martingale deflators for a large class of initial models (i.e.,models in the small filtration) that are already risk-neutralized.

In addition to its further exploration of the subject of peacocks, introduced in recent Séminaires de Probabilités, this volume continues the series’ focus on current research themes in traditional topics such as stochastic calculus, filtrations and random matrices. Also included are some particularly interesting articles involving harmonic measures, random fields and loop soups. The featured contributors are Mathias Beiglböck, Martin Huesmann and Florian Stebegg, Nicolas Juillet, Gilles Pags, Dai Taguchi, Alexis Devulder, Mátyás Barczy and Peter Kern, I. Bailleul, Jürgen Angst and Camille Tardif, Nicolas Privault, Anita Behme, Alexander Lindner and Makoto Maejima, Cédric Lecouvey and Kilian Raschel, Christophe Profeta and Thomas Simon, O. Khorunzhiy and Songzi Li, Franck Maunoury, Stéphane Laurent, Anna Aksamit and Libo Li, David Applebaum, and Wendelin Werner. .

The paper gathers together ideas related to thin random time, i.e., random time whose graph is contained in a thin set. The concept naturally completes the studies of random times and progressive enlargement of filtrations. We develop classification and (*)-decomposition of random times, which is analogous to the decomposition of a stopping time into totally inaccessible and accessible parts, and we show applications to the hypothesis (H ′), honest times and informational drift via entropy.

This volume is dedicated to the memory of Marc Yor, who passed away in 2014. The invited contributions by his collaborators and former students bear testament to the value and diversity of his work and of his research focus, which covered broad areas of probability theory. The volume also provides personal recollections about him, and an article on his essential role concerning the Doeblin documents. With contributions by P. Salminen, J-Y. Yen & M. Yor. J. Warren. T. Funaki. J. Pitman& W. Tang. J-F.

We study problems related to the predictable representation property for a progressive enlargement G of a reference filtration F through observation of a finite random time τ. We focus on cases where the avoidance property and/or the continuity property for F-martingales do not hold and the reference filtration is generated by a Poisson process. Our goal is to find out whether the predictable representation property (PRP), which is known to hold in the Poisson filtration, remains valid for a progressively enlarged filtration G with respect to a judicious choice of G-martingales.

Given a reference filtration F, we consider the cases where an enlarged filtration G is constructed from 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 G-optional semimartingale decomposition for F-local martingales. Our study is then applied to answer 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. We provide alternative proofs of some results from [5], with a methodology based on our optional semimartingale decomposition, which reduces significantly the length of the proof.

This thesis deals with problems associated with the theory of filtration magnification. It is divided into two parts: the first part is devoted to random times. We study the properties of the different classes of random times from the point of view of the filtration magnification. The second part concerns the study of the stability of the arbitration condition on the filtration magnification.

This paper quantifies the impact of stopping at a random time on non-arbitrage, for a class of semimartingale models. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, also known in the literature as the arbitrage of the first kind. The first principal result lies in describing the pairs of market model and random times for which the resulting stopped model fulfills the NUPBR condition. The second principal result characterises the random time models that preserve the NUPBR property after stopping for any quasi-left-continuous market model. The analysis that drives these results is based on new stochastic developments in martingale theory with progressive enlargement of filtration. Furthermore, we construct explicit martingale densities (deflators) for a subclass of local martingales when stopped at a random time.

This thesis treats the problems settled in enlargement of filtraion theory. It consists of two parts. The first part is devoted to random times. We study the properties of different classes of random times from enlargement of filtration point of view. The second part concerns the study of the stability of the no arbitrage condition under enlargement of filtration. We are mainly interested in No Unbounded Profit with Bounded Risk condition. We study absence of arbitrage in the case of progressive enlargement of up to random time. Then we look at the case of initial enlargement with random variable satisfying Jacod's hypothesis.