Trees and asymptotic developments for fractional stochastic differential equations A. Neuenkirch?, I. Nourdin†, A. Roßler‡ and S. Tindel November 10, 2006 Abstract 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 [10], we show how to obtain 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. With respect to [2], 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. Keywords: fractional Brownian motion, stochastic differential equations, trees expan- sions. MSC: 60H05, 60H07, 60G15 1 Introduction In this article, we study the stochastic differential equation (SDE in short) Xat = a + ∫ t 0 ?(Xas )dBs + ∫ t 0 b(Xas )ds, t ? [0, T ], (1) ?Johann Wolfgang Goethe-Universitat Frankfurt am Main, FB Informatik und Mathematik, Robert- Mayer-Strasse 10, 60325 Frankfurt am Main, Germany, neuenkir@math.
- young integral
- lts
- skorohod stochastic
- stochastic differential
- hurst parameter
- all monotonically labelled
- newton-cotes integral corrected
- differential equation driven
- trees having