The theme of providing predictable criteria for absolute continuity and for mutual singularity of two density processes on a filtered probability space is extensively studied, e.g., in the monograph by J. Jacod and A. N. Shiryaev [JS]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open ([JS], p210): "We do not know whether it is possible to derive a predictable criterion (necessary and sufficient condition) for "P'T..." (expression not representable in this abstract). It turns out that to this question raised in [JS] which we also chose as the title of this note, there are two answers: on the negative side we give an easy example, showing that in general the answer is no, even when we use a rather wide interpretation of the concept of "predictable criterion". The difficulty comes from the fact that the density process of a probability measure P with respect to another measure P' may suddenly jump to zero. On the positive side we can characterize the set, where P' becomes singular with respect to P - provided this does not happen in a sudden but rather in a continuous way - as the set where the Hellinger process diverges, which certainly is a "predictable criterion". This theorem extends results in the book of J. Jacod and A. N. Shiryaev [JS]. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

We consider transformations of a given random walk on a countable group determined by Markov stopping times. We prove that these transformations preserve the Poisson boundary. Moreover, under some mild conditions, the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory.

The most important market segmentation in sports and entertainment industry is the competition between customers that buy bundle and single tickets. A common selling practice is starting the selling season with bundle ticket sales and switching to selling single tickets later on. The aim of this practice is to increase the number of customers that buy bundles, to create a fund before the season starts and to increase the load factor of the games with low demand. In this thesis, we investigate the effect of time dependent demand on dynamic switching times and the potential revenue gain over the case where the demand rate is assumed to be constant with time.

Ο σκοπός αυτής της διπλωματικής εργασίας είναι να συζητηθούν και να παρουσιαστούν οι υπάρχουσες τεχνικές που εφαρμόζουν την στρατηγική δειγματοληψίας βασισμένη σε γεγονότα για την εκτίμηση της κατάστασης των γραμμικών συστημάτων και να αξιολογηθεί πώς αυτές οι μέθοδοι επηρεάζουν τη συνολική απόδοση του συστήματος. Όπως ήδη αναφέρθηκε, περιορισμοί στον ρυθμό δειγματοληψίας ανακύπτουν στα NCS λόγω του περιορισμένου διαθέσιμου εύρους ζώνης. Αυτός ο περιορισμός στον αριθμό δειγμάτων επηρεάζει σαφώς το μέσο τετραγωνικό σφάλμα εκτίμησης, και η ερώτηση είναι πώς πρέπει να επιλέξουμε τις χρονικές στιγμές δειγματοληψίας προκειμένου να το ελαχιστοποιήσουμε. / The aim of this diploma thesis is to discuss and present existing techniques that apply event-triggered sampling to linear system state estimation, and to evaluate how these methods affect the overall performance of the system. As already mentioned, sampling rate constraints arise in NCS due to the limited bandwidth available. This restriction on the number of samples clearly affects the mean square estimation distortion error, and the question is how we should choose the sampling instants in order to minimize it.

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This thesis consists of extensions of results on a perpetual American swaption problem. Companies routinely plan to swap uncertain benefits with uncertain costs in the future for their own benefits. Our work explores the choice of timing policies associated with the swap in the form of an optimal stopping problem. In this thesis, we have shown that Hu, Oksendal's (1998) condition given in their paper to guarantee that the optimal stopping time is a.s. finite is in fact both a necessary and sufficient condition. We have extended the solution to the problem from a region in the parameter space where optimal stopping times are a.s. finite to a region where optimal stopping times are non-a.s. finite, and have successfully calculated the probability of never stopping in this latter region. We have identified the joint distribution for stopping times and stopping locations in both the a.s. and non-a.s. finite stopping cases. We have also come up with an integral formula for the inner product of a generalized hyperbolic distribution with the Cauchy distribution. Also, we have applied our results to a back-end forestry harvesting model where stochastic costs are assumed to exponentiate upwards to infinity through time. / Graduation date: 2013

Eine Klasse von Markovprozessen mit deterministischem Pfaden und zufälligen Sprüngen wird unter Zuhilfenahme von Martingalen und des erweiterten infinitesimalen Generators untersucht. Dabei steht die Berechnung des Erwartungswertes und der Laplacetransformierten bestimmter Stoppzeiten im Vordergrund. Des weiteren wird die Frage untersucht, wann die in Frage kommenden Prozesse über stationäre Verteilungen verfügen und wie diese im Existenzfall beschaffen sind. Die Methoden werden am Beispiel eines verallgemeinerten Altersprozesses und eines Lager- bzw. Dammprozesses vorgeführt.

Cette thèse est composée de deux parties indépendantes : la première partie traite des équations différentielles stochastiques dans le cadre de la G-espérance, tandis que la deuxième partie présente les résultats obtenus pour les équations différentielles stochastiques du seconde ordre. Dans un premier temps, on considère les intégrales stochastiques par rapport à un processus croissant, et on donne une extension de la formule d'Itô dans le cadre de la G-espérance. Ensuite, on étudie une classe d'équations différentielles stochastiques réfléchies unidimensionnelles dirigées par un G-mouvement brownien. Dans la suite, en utilisant une méthode de localisation, on prouve l'existence et l'unicité de solutions pour les équations différentielles stochastiques dirigées par un G-mouvement brownien, dont les coefficients sont localement lipschitziens. Enfin, dans le même cadre, on discute des problèmes de réflexion multidimensionnelle et on fournit quelques résultats de convergence. Dans un deuxième temps, on étudie une classe d'équations différentielles stochastiques rétrogrades du seconde ordre à croissance quadratique. Le but de ce travail est de généraliser le résultat obtenu par Possamaï et Zhou en 2012. On montre aussi l'existence et l'unicité des solutions pour ces équations, mais sous des hypothèses plus faibles. De plus, ce résultat théorique est appliqué aux problèmes de maximisation robuste de l'utilité du portefeuille en finance. / This thesis consists of two relatively independent parts : the first part concerns stochastic differential equations in the framework of the G-expectation, while the second part deals with a class of second order backward stochastic differential equations. In the first part, we first consider stochastic integrals with respect to an increasing process and give an extension of Itô's formula in the G-framework. Then, we study a class of scalar valued reflected stochastic differential equations driven by G-Brownian motion. Subsequently, we prove the existence and the uniqueness of solutions for some locally Lipschitz stochastic differential equations driven by G-Brownian motion. At the end of this part, we consider multidimensional reflected problems in the G-framework, and some convergence results are obtained. In the second part, we study the wellposedness of a class of second order backward stochastic differential equations (2BSDEs) under a quadratic growth condition on their coefficients. The aim of this part is to generalize a wellposedness result for quadratic 2BSDEs by Possamaï and Zhou in 2012. In this thesis, we work under some usual assumptions and deduce the existence and uniqueness theorem as well. Moreover, this theoretical result for quadratic 2BSDEs is applied to solve some robust utility maximization problems in finance.