A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index $H&lt;1/4$

Unterberger, Jeremie

doi:10.48550/arxiv.0808.3458

Cited by 3 publications

(3 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Despite an abstract (non constructive) proof of existence [36], and several recent investigations [51,50,14] yielding a sort of general classification of rough paths in the algebraic sense, this series of papers gives the first construction of a rough path over B for α ≤ 1/4 by means of an explicit sequence of approximations. The barrier at α = 1/4 has been recognized by several authors using different approaches [11,42,47,48], and shown to extend to other models as well [26].…”

Section: Introductionmentioning

confidence: 85%

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

Magnen

Unterberger

2011

Ann. Henri Poincaré

View full text Add to dashboard Cite

Let B = (B1(t), . . . , B d (t)) be a d-dimensional fractional Brownian motion with Hurst index α < 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular nonGaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization.

show abstract

Section: Introductionmentioning

confidence: 85%

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

Magnen

Unterberger

2011

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…une intégrale itérée d'ordre 2, définie comme limite des intégrales itérées des interpolations linéaires par morceaux des trajectoires, diverge quand α ≤ 1/4. Des travaux ultérieurs reposant sur des méthodes différentes [15,57,65,66] ont confirmé l'existence de cette barrière apparemment infranchissable en α = 1/4. Et pourtant, la théorie des chemins rugueux * (ou rough paths), une théorie d'intégration adaptée aux chemins irréguliers, introduite par T. Lyons à la fin des années 90 [46,47] et devenue un outil essentiel en calcul stochastique [46,47,33,44,45,25], prédit -malheureusement par des arguments géométriques non constructifs -l'existence d'approximations C 1 par morceaux, autres que l'interpolation linéaire par morceaux, dont les intégrales itérées de tous ordres convergent vers des quantités finies s'interprétant comme substituts d'intégrales itérées du brownien fractionnaire -en termes plus géométriques, comme chemin rugueux au-dessus du brownien -.…”

Section: Introductionunclassified

Mode d'emploi de la théorie constructive des champs bosoniques (A user's guide to bosonic constructive field theory)

Unterberger

2011

Preprint

View full text Add to dashboard Cite

Mode d'emploi de la théorie constructive des champs bosoniques avec une application aux chemins rugueux Jérémie Unterberger Nous développons dans cet article les principaux arguments constructifs utilisés en théorie quantique des champs, en nous cantonnant aux théories bosoniques, pour lesquelles il n'existe pas de présentation générale récente. L'article s'adresse d'abord et avant tout à des mathématiciens ou physiciens mathématiciens connaissant les arguments de base de la théorie perturbative des champs, et souhaitant connaître un cadre général dans lequel ils peuvent être rendus rigoureux. Il fournit également un aperçu d'une série d'articles récents [50, 51] visant à donner une définition constructive des chemins rugueux et du calcul stochastique fractionnaire.We develop in this article the principal constructive arguments used in quantum field theory, limiting us to bosonic theories, for which there does not exist any recent general presentation. The article is primarily written for mathematicians or mathematical physicists knowing the basic arguments of quantum field theory, and desiring to discover a general framework in which they can be made rigorous. It also provides a glimpse of a recent series of articles [50,51] whose aim is to give a constructive definition of rough paths and fractionary stochastic calculus.Mots clé: théorie constructive des champs, renormalisation, développements cluster, Brownien fractionnaire, calcul stochastique fractionnaire, chemins rugueux.

show abstract

“…called Lévy area. The corresponding Stratonovich integral, obtained as a limit either by linear interpolation or by more refined Gaussian approximations [11,51,58,59], has been shown to diverge as soon as α ≤ 1/4. This seemingly no-go theorem, although clear and derived by straightforward computations that we reproduce in short in section 1, appears to be a puzzle when put in front of the results of rough path theory [43,44,28,39,40,20].…”

Section: Introductionmentioning

confidence: 99%

From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics

Magnen,

Unterberger

2010

Preprint

View full text Add to dashboard Cite

Let B = (B1(t), . . . , B d (t)) be a d-dimensional fractional Brownian motion with Hurst index α ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B.We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read in a second time the companion article [48] (or a preliminary version [47]) for the constructive proofs.

show abstract

A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index $H<1/4$

Cited by 3 publications

References 17 publications

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

Mode d'emploi de la théorie constructive des champs bosoniques (A user's guide to bosonic constructive field theory)

From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics

Contact Info

Product

Resources

About