Stochastic Calculus with respect to Gaussian Processes

Stochastic integration with respect to Gaussian processes, such as fractional Brownian motion (fBm) or multifractional Brownian motion (mBm), has raised strong interest in recent years, motivated in particular by applications in finance, Internet traffic modeling and biomedicine. The aim of this work to define and develop, using White Noise Theory, an anticipative stochastic calculus with respect to a large class of Gaussian processes, denoted G, that contains, among many other processes, Volterra processes (and thus fBm) and also mBm. This stochastic calculus includes a definition of a stochastic integral, It\^o formulas (both for tempered distributions and for functions with sub-exponential growth), a Tanaka Formula as well as a definition, and a short study, of (both weighted and non weighted) local times of elements of G . In that view, a white noise derivative of any Gaussian process G of G is defined and used to integrate, with respect to G, a large class of stochastic processes, using Wick products. A comparison of our integral wrt elements of G to the ones provided by Malliavin calculus in [AMN01] and by It\^o stochastic calculus is also made. Moreover, one shows that the stochastic calculus with respect to Gaussian processes provided in this work generalizes the stochastic calculus originally proposed for fBm in [EVdH03, BS{\O}W04, Ben03a] and for mBm in [LLV14, Leb13, LLVH14]. Likewise, it generalizes results given in [NT06] and some results given in [AMN01]. In addition, it offers alternative conditions to the ones required in [AMN01] when one deals with stochastic calculus with respect to Gaussian processes.Comment: (26/07/2014). Previously this work appeared as arXiv:1703.08393 which was incorrectly submitted as a new paper (and has now been withdrawn

White noise-based stochastic calculus with respect to multifractional Brownian motion

International audienceStochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is a Gaussian extension of fBm that allows to control the pointwise regularity of the paths of the process and to decouple it from its long range dependence properties. This generalization is obtained by replacing the constant Hurst parameter H of fBm by a function h(t). Multifractional Brownian motion has proved useful in many applications, including the ones just mentioned. In this work we extend to mBm the construction of a stochastic integral with respect to fBm. This stochastic integral is based on white noise theory, as originally proposed in [15], [6], [4] and in [5]. In that view, a multifractional white noise is defined, which allows to integrate with respect to mBm a large class of stochastic processes using Wick products. Itô formulas (both for tempered distributions and for functions with sub-exponential growth) are given, along with a Tanaka Formula. The cases of two specific functions h which give notable Itô formulas are presented. We also study a stochastic differential equation driven by a mBm. Keywords: multifractional Brownian motion, Gaussian processes, White Noise Theory, S-Transform, Wick-Itô integral, Itô formula, Tanaka formula, Stochastic differential equations

Pain management procedures used by dental and maxillofacial surgeons: an investigation with special regard to odontalgia

BACKGROUND: Little is known about the procedures used by German dental and maxillofacial surgeons treating patients suffering from chronic orofacial pain (COP). This study aimed to evaluate the ambulatory management of COP. METHODS: Using a standardized questionnaire we collected data of dental and maxillofacial surgeons treating patients with COP. Therapists described variables as patients' demographics, chronic pain disorders and their aetiologies, own diagnostic and treatment principles during a period of 3 months. RESULTS: Although only 13.5% of the 520 addressed therapists returned completely evaluable questionnaires, 985 patients with COP could be identified. An orofacial pain syndrome named atypical odontalgia (17.0 %) was frequent. Although those patients revealed signs of chronification, pain therapists were rarely involved (12.5%). For assessing pain the use of Analogue Scales (7%) or interventional diagnostics (4.6%) was uncommon. Despite the fact that surgical procedures are cofactors of COP therapists preferred further surgery (41.9%) and neglected the prescription of analgesics (15.7%). However, most therapists self-evaluated the efficacy of their pain management as good (69.7 %). CONCLUSION: Often ambulatory dental and maxillofacial surgeons do not follow guidelines for COP management despite a high prevalence of severe orofacial pain syndromes

Calcul stochastique par rapport au mouvement brownien multifractionnaire et applications à la finance

The aim of this PhD Thesis was to build and develop a stochastic calculus (in particular a stochastic integral) with respect to multifractional Brownian motion (mBm). Since the choice of the theory and the tools to use was not fixed a priori, we chose the White Noise theory which generalizes, in the case of fractional Brownian motion (fBm) , the Malliavin calculus. The first chapter of this thesis presents several notions we will use in the sequel.In the second chapter we present a construction as well as the main properties of stochastic integral with respect to harmonizable mBm.We also give Ito formulas and a Tanaka formula with respect to this mBm. In the third chapter we give a new definition, simplier and generalier of multifractional Brownian motion. We then show that mBm appears naturally as a limit of a sequence of fractional Brownian motions of different Hurst index.We then use this idea to build an integral with respect to mBm as a limit of sum of integrals with respect ot fBm. This being done we particularize this definition to the case of Malliavin calculus and White Noise theory. In this last case we compare the integral hence defined to the one we got in chapter 2. The fourth and last chapter propose a multifractional stochastic volatility model where the process of volatility is driven by a mBm. The interest lies in the fact that we can hence take into account, in the same time, the long range dependence of increments of volatility process and the fact that regularity vary along the time.Using the functional quantization theory in order to, among other things, approximate the solution of stochastic differential equations, we can compute the price of forward start options and then get and plot the implied volatility nappe that we graphically represent.Le premier chapitre de cette thèse introduit les différentes notions que nous utiliserons et présente les travaux qui constituent ce mémoire.Dans le deuxième chapitre de cette thèse nous donnons une construction ainsi que les principales propriétés de l'intégrale stochastique par rapport au mBm harmonisable. Y sont également établies des formules d'Itô et une formule de Tanaka pour l'intégrale stochastique par rapport à ce mBm..Dans le troisième chapitre nous donnons une nouvelle définition, à la fois plus simple et plus générale, du mouvement brownien multifractionnaire. Nous montrons ensuite que le mBm apparaît naturellement comme limite de suite de somme de mouvement brownien fractionnaire (fBm) d’indices de Hurst différents.Nous appliquons alors cette idée pour tenter de construire une intégrale stochastique par rapport au mouvement brownien multifractionnaire à partir d’intégrales par rapport au fBm. Cela fait nous appliquons cette définition d’intégrale par rapport au mBm pour une méthode d’intégration donnée aux deux méthodes que sont le calcul de Malliavin et la théorie du bruit blanc.Dans ce dernier cas nous comparons alors l’intégrale ainsi construite à celle obtenue au chapitre 2. Le quatrième et dernier chapitre est une application du calcul stochastique développé dans les chapitres précédents. Nous y proposons un modèle à volatilité multifractionnaire où le processus de volatilité est dirigée par un mBm. L’intérêt résidant dans le fait que l’on peut ainsi prendre en compte à la fois la dépendance à long terme des accroissements de la volatilité mais aussi le fait que la trajectoire de ces accroissements varie au cours du temps.Utilisant alors la théorie de la quantification fonctionnelle pour, entre autres, approximer la solution de certaines des équations différentielles stochastiques, nous parvenons à calculer le prix d’option à départ forward et implicitons ainsi une nappe de volatilité que l’on représente graphiquement pour différentes maturités

Local Times of Gaussian Processes: Stochastic Calculus with respect to Gaussian Processes Part II

The aim of this work is to define and perform a study of local times of all Gaussian processes that have an integral representation over a real interval (that maybe infinite). Very rich, this class of Gaussian processes, contains Volterra processes (and thus fractional Brownian motion), multifractional Brownian motions as well as processes, the regularity of which varies along the time. Using the White Noise-based anticipative stochastic calculus with respect to Gaussian processes developed in [Leb17], we first establish a Tanaka formula. This allows us to define both weighted and non-weighted local times and finally to provide occupation time formulas for both these local times. A complete comparison of the Tanaka formula as well as the results on Gaussian local times we present here, is made with the ones proposed in [MV05, LN12, SV14]