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 $$ \mathrm{E}[Δ_hf(X_t)], \mathrm{E}[Δ_hf(X_t)|ℱ_t^X] \mathrm{and} \mathrm{E}[Δ_hf(X_t)|X_t], $$ where $Δ_hf(X_t)=f(X_{t+h})−f(X_t)$, 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.