Trees and asymptotic expansions for fractional stochastic differential equations

Andreas Neuenkirch (1) , Ivan Nourdin (2) , A. Rössler (3) , Samy Tindel (2, 4)

1 FB Informatik und Mathematik
2 IECN - Institut Élie Cartan de Nancy
3 Department of Mathematics [Darmstadt]
4 TOSCA - Simuler et calibrer des modèles stochastiques

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.

Mots clés: Fractional Brownian motion; Stochastic differential equations; Trees expansions

Domaines: Probabilités [math.PR]

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, 2009, 45 (1), pp.157-174. ⟨10.1214/07-AIHP159⟩. ⟨hal-00602404⟩