# Calcul stochastique généralisé et applications au mouvement brownien fractionnaire ; Estimation non-paramétrique de la volatilité et test d'adéquation

Abstract : In the first part of this thesis, we introduce the notion of m-order integrals and we associate with some Ito's type formulae. The case of fractional Brownian motion is investigated. In the second part, we study first and second orders approximations of m-order integrals, the convergence being almost surely or in law. In the third part, we study differential equations driven by an Holder continuous function. We prove existence and uniqueness and we give two approximations of the solution. In the fourth part, we prove that under suitable assumptions, the solution at time $t$ to a stochastic differential equation driven by a fractional Brownian motion, has a density. The last part deals with estimation of the volatility coefficient in a classical model. We also construct a goodness-of-fit test.
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https://tel.archives-ouvertes.fr/tel-01754362
Contributor : Ivan Nourdin <>
Submitted on : Monday, February 28, 2005 - 9:02:16 PM
Last modification on : Thursday, January 17, 2019 - 5:20:19 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:09:57 PM

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• HAL Id : tel-01754362, version 2

### Citation

Ivan Nourdin. Calcul stochastique généralisé et applications au mouvement brownien fractionnaire ; Estimation non-paramétrique de la volatilité et test d'adéquation. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2004. Français. ⟨NNT : 2004NAN10040⟩. ⟨tel-01754362v2⟩

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