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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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Part of the Lecture Notes in Mathematics book series (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.

Key words

  • Stochastic differential equation
  • fractional Brownian motion
  • Russo-Vallois integrals
  • Newton-Cotes functional
  • Approximation schemes
  • Doss-Sussmann transformation

MSC 2000

  • 60G18
  • 60H05
  • 60H20

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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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