Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on Zd{\Bbb Z}^d

Using a method developed by Durrett and Resnick [22] we establish general criteria for the convergence of properly rescaled clock processes of random dynamics in random environments on infinite graphs. This complements the results of [26], [19], and [20]: put together these results provide a unified framework for proving convergence of clock processes. As a first application we prove that Bouchaud's asymmetric trap model on ${\Bbb Z}^d$ exhibits a normal aging behavior for all $d\geq 2$. Namely, we show that certain two-time correlation functions, among which the classical probability to find the process at the same site at two time points, converge, as the age of the process diverges, to the distribution function of the arcsine law. As a byproduct we prove that the fractional kinetics process ages

Time consistency of multi-period distortion measures

Dynamic risk measures play an important role for the acceptance or non-acceptance of risks in a bank portfolio. Dynamic consistency and weaker versions like conditional and sequential consistency guarantee that acceptability decisions remain consistent in time. An important set of static risk measures are so-called distortion measures. We extend these risk measures to a dynamic setting within the framework of the notions of consistency as above. As a prominent example, we present the Tail-Value-at-Risk (TVaR

Contribution to the study of aging in disordered systems

Nous étudions mécanismes généraux qui sont à l'origine de vieillissement de dynamiques en environnements aléatoires, connu sous. Le vieillissement s'observe dans le comportement de certaines fonctions de corrélation, qui ne deviennent jamais indépendantes de l'âge du système. Une approche universelle à ce problème fut développée durant les dernières décennies: le comportement des fonctions de corrélation peut être lié à celui du processus d'horloge, qui est le temps total écoulé le long d'une trajectoire de la dynamique.Une approche élégante fut proposée par Gayrard (2010, 2012) pour étudier le processus d'horloge. Celui-ci est vu comme un processus de sommes partielles à incréments corrélés auquel des critères de convergence, dûs à Durett et Resnick (1978) sont appliqués. Cette méthode fut poussée plus avant par Bovier et Gayrard (2013).Nous étendons les méthodes développées par Gayrard (2012) et Bovier et Gayrard (2013), et étudions vieillissement dans divers modèles. Dans la première partie, nous établissons des critères de convergence vers des processus extrémaux pour des graphes finis et improuver résulats obtenus par Ben Arous et Gun (2012) sur le vieillissement extrémal. La deuxième partie traite de dynamiques sur des graphes infinis. Nous donnons des conditions suffisantes sous lesquelles le processus d'horloge sous-jacent converge vers un subordinateur, et établir l'existence de vieillissement normal dans le modèle assymétrique de pièges de Bouchaud sur Z^d pour d≥2. La troisième partie concerne le modèle de Bouchaud assymétrique lorsque d≥3 et sa version symétrique lorsque d=2. Nous prouvons l'existence d'un régime de sur-vieillissement.We study general mechanisms that lead to aging behavior of dynamics in random environments. Aging is observed in the behavior of correlation functions that never become independent of the age of the system. A universal approach to this problem was developed over the past decades: the behavior correlation functions can be linked to the long-time behavior of the clock process, which is the total time elapsed along the trajectory of the random motion. An elegant approach to studying clock processes was proposed by Gayrard (2010,2012). Here, the clock process is viewed as a partial sum process whose increments are dependent random variables and then convergence criteria, due to Durrett and Resnick (1978), are employed. This method was further developed by Bovier and Gayrard (2013).We extend the methods of Gayrard (2012) and Bovier and Gayrard (2013) and use our methods to study the aging behavior of various models. In the first part we establish criteria for the convergence of clock processes on sequences of finite graphs to extremal processes and improve results on extremal aging obtained by Ben Arous and Gun (2012). The second part deals with dynamics that are defined on infinite graphs. We introduce sufficient conditions for the clock process to converge to a subordinator and establish the existence of a normal aging regime in Bouchaud's asymmetric trap model on Z^d, for d≥2. In the third part of this thesis we consider Bouchaud's asymmetric trap model for d≥3, and its symmetric version for d=2. We prove the existence of an super-aging regime

Time consistency of multi-period distortion measures

ISSN:0721-2631ISSN:2193-140