Trees and asymptotic developments for fractional stochastic differential equations

Andreas Neuenkirch (1) , Ivan Nourdin (2) , Andreas Roessler (3) , Samy Tindel (4)

1 FB Informatik und Mathematik
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 Department of Mathematics [Darmstadt]
4 IECN - Institut Élie Cartan de Nancy

In this paper we consider a n-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H>1/3. After solving this equation in a rather elementary way, following the approach of Gubinelli, we show how to obtain an expansion for E[f(X_t)] in terms of t, where X denotes the solution to the SDE and f:R^n->R is a regular function. With respect to the work by Baudoin and Coutin, where the same kind of problem is considered, we try an improvement in three different directions: we are able to take a drift into account in the equation, we parametrize our expansion with trees (which makes it easier to use), and we obtain a sharp control of the remainder.

Mots clés: fractional Brownian motion; stochastic differential equations; trees expansions.; trees expansions

Domaines: Probabilités [math.PR]

Andreas Neuenkirch, Ivan Nourdin, Andreas Roessler, Samy Tindel. Trees and asymptotic developments for fractional stochastic differential equations. 2006. ⟨hal-00112928⟩