Asymptotic expansions at any time for scalar fractional SDEs with Hurst index H> 1/2

Abstract

We study the asymptotic expansions with respect to h of E[∆hf(Xt)], E[∆hf(Xt)|F X t] and E[∆hf(Xt)|Xt], where ∆hf(Xt) = f(Xt+h) − f(Xt), when f:R → R is a smooth real function, t ≥ 0 is a fixed time, X is the solution of a one-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst index H> 1/2 and F X is its natural filtration