Trees and asymptotic expansions for fractional stochastic differential equations

Andreas Neuenkirch 1 Ivan Nourdin 2 A. Rößler 3 Samy Tindel 2, 4
4 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : In this article, we consider an n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameterH >1/3. We derive an expansion for E[f (Xt )] in terms of t, where X denotes the solution to the SDE and f :Rn →R is a regular function. Comparing to F. Baudoin and L. Coutin, Stochastic Process. Appl. 117 (2007) 550-574, where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case H >1/2.
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Submitted on : Wednesday, June 22, 2011 - 2:00:54 PM
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Andreas Neuenkirch, Ivan Nourdin, A. Rößler, Samy Tindel. Trees and asymptotic expansions for fractional stochastic differential equations. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2009, 45 (1), pp.157-174. ⟨10.1214/07-AIHP159⟩. ⟨hal-00602404⟩

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