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A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one

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Séminaire de Probabilités XLI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1934))

Abstract

We will focus — in dimension one — on the SDEs of the type dX t = σ(X t )dB t + b(X t )dt where B is a fractional Brownian motion. Our principal aim is to describe a simple theory — from our point of view — allowing to study this SDE, and this for any H∈(0,1). We will consider several definitions of solutions and, for each of them, study conditions under which one has existence and/or uniqueness. Finally, we will examine whether or not the canonical scheme associated to our SDE converges, when the integral with respect to fBm is defined using the Russo-Vallois synmetric integral.

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie Paris VI, Boîte courrier 188, 4 Place Jussien, 75252, Paris Cedex 5, France

    Ivan Nourdin

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  1. Ivan Nourdin
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Editors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boîte courrier I88 4, place Jussieu, 75252, Paris cedex 05, France

    Catherine Donati-Martin

  2. Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7, rue René Descartes, 67084, Strasbourg cedex, France

    Michel Émery

  3. Laboratoire de Mathématiques Bâtiment Fermat, Université Versailles-Saint-Quentin, 45, avenue des Etats-Unis, 78035, Versailles cedex, France

    Alain Rouault

  4. UFR Sciences et techniques, Université de Besançon, 16, route de Gray, 25030, Besançon cedex, France

    Christophe Stricker

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Nourdin, I. (2008). A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_8

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