Filtered Brownian motions as weak limit of filtered Poisson processes

International audienceThe main result of this paper is a limit theorem which shows the convergence in law, on a Hölderian space, of filtered Poisson processes (a class of processes which contains shot noise process) to filtered Brownian motion (a class of processes which contains fractional Brownian motion) when the intensity of the underlying Poisson process is increasing. We apply the theory of convergence of Hilbert space valued semi-martingales and use some result of radonification

On Ornstein-Uhlenbeck driven by Ornstein-Uhlenbeck processes

International audienceWe investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein-Uhlenbeck processes driven by Ornstein-Uhlenbeck processes

Mouvement Brownien Fractionnaire, applications aux télécommunications. Calcul Stochastique relativement à des processus fractionnaires.

Président : Jean Mémin, Rapporteurs : David Nualart et Nicolas Privault, Examinateurs : Laurent Decreusefond et Ying Hu.The fractional Brownian motion (fBm) has become a key process as soon as one wants to free oneself from the Markov and independence of increments properties. We have given the main properties of this process and we have insisted on certain aspects of its use as fluid queue model. Then, we have developed construction of an anticipative integral with respect to fBm from an anticipative integral with respect to the Brownian motion. Then, we have introduced an anticipative integral with respect to filtered Poisson process (fPp) from an anticipative integral with respect to marked Poisson process, an integral that we have connected to the Stieltjès integral. Our study has gone on with an Itô formula for cylindrical functional and a Hölder continuity theorem for integrated processes. To conclude, a weak convergence theorem for a sequence of fPp to a Volterra Process has been established.Le mouvement Brownien fractionnaire (mBf) est devenu un processus incontournable dès que l'on veut s'affranchir des propriétés de Markov et d'indépendance des accroissements. Nous verrons les principales propriétés de ce processus, nous insisterons sur certains aspects de son utilisation comme modèle de file fluide. On développe ensuite la construction d'une intégrale anticipative relative au mBf à partir de l'intégrale anticipative relative au mouvement Brownien. Fort de cette idée, nous avons introduit une intégrale anticipative relative à des processus de Poissons filtrés (pPf) à partir d'une intégrale anticipative pour des processus de Poissons marqués, intégrale que nous relions à l'intégrale de Stieltjès. L'étude se poursuit par une formule de Itô pour des fonctionnelles cylindriques et par un résultat sur la continuité de Holdër des processus intégrés. Pour finir, un théorème de convergence en loi d'une suite de pPf vers un processus de Volterra est établi

Upper bounds on Rubinstein distances on configuration spaces and applications

In this paper, we provide upper bounds on several Rubinstein-type distances on the configuration space equipped with the Poisson measure. Our inequalities involve the two well-known gradients, in the sense of Malliavin calculus, which can be defined on this space. Actually, we show that depending on the distance between configurations which is considered, it is one gradient or the other which is the most effective. Some applications to distance estimates between Poisson and other more sophisticated processes are also provided, and an application of our results to tail and isoperimetric estimates completes this work.Comment: To appear in Communications on Stochastic Analysis and Application

Statistique des processus - Application en Finance

National audienceLe développement et l'analyse de modèles financiers et des objets qui lui sont associés ont attiré une grande attention ces dernières années. En effet, ces concepts offrent une grande complexité de modélisation et, de fait, des modèles de plus en plus compliqués ont été développés. Le calcul stochastique a été largement utilisé, à commencer par la fameuse formule de Black et Scholes, tout en étant fortement décrié du fait de la crise financière. De fait, le calcul stochastique peut aider à comprendre les phénomènes en les modélisant. Mais, une fois ces modèles mis en place, une nouvelle difficulté apparaît, celle de la calibration à savoir comment estimer à partir de données réelles les différents paramètres d'un modèle. Dans les exposés de cette session nous montrerons la grande diversité des domaines d'application du calcul stochastique en Finance. L'accent sera mis sur l'aspect statistique de ces problèmes

Quantifying the combined effects of the heating time, the temperature and the recovery medium pH on the regrowth lag time of Bacillus cereus spores after a heat treatment

International audienceThe purpose of this study was to quantify the lag time of re-growth of heated spores of Bacillus cereus as a function of the conditions of the heat treatment: temperature, duration and pH of the recovery medium. For a given heating temperature, curves plotting lag times versus time of heating show more or less complex patterns. However, under a heating time corresponding to a decrease of 2 decimal logarithms of the surviving populations of spores, a linear relationship between the lag time of growth and the time of the previous heat treatment can be observed. The slope of this linear relationship followed itself a Bigelow type linear relationship, the slope of which yielded a - value very close to the observed conventional z-value. It was then concluded that the slope of the regrowth lag time versus the heating time followed a linear relationship with the sterilisation value reached in the course of the previous heat treatment. A sharp effect of the pH of the medium which could be described by a simple "secondary" model was observed. As expected, the observed intercept of the linear relationship between lag time and heating time (lag without previous heating) was dependent on only the pH of the medium and not on the heating temperature

Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition

A filtered process $X^k$ is defined as an integral of a deterministic kernel $k$ with respect to a stochastic process $X$. One of the main problems to deal with such processes is to define a stochastic integral with respect to them. When $X$ is a Brownian motion one can use the Gaussian properties of $X^k$ to define an integral intrinsically. When $X$ is a jump process or a Levy process, this is not possible. Alternatively, we can use the integrals defined by means of the so called $\mathcal{S}$-transform or by means of the integral with respect to the process $X$ and a linear operator $\mathcal{K}$ constructed from $k$. The usual fact that even for predictable $Y$, $K^{\ast}(Y)$ may not be predictable forces us to consider only anticipative integrals. The aim of this paper is, on the one hand, to clarify the links between these integrals for a given $X$ and on the other hand, to investigate how the Lévy-Itô decomposition of a Levy process $L$, roughly speaking $L=B+J$, where $B$ is a Brownian motion and $J$ is a pure jump Lévy process, behaves with respect to these integrals