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 \[\mathrm{E}[\Delta_hf(X_t)],\qquad \mathrm{E}[\Delta_hf(X_t)|\mathscr{F}^X_t]\quadand\quad \mathrm{E}[\Delta_hf(X_t)|X_t],\] where $\Delta_hf(X_t)=f(X_{t+h})-f(X_t)$, when $f:\mathbb {R}\to\mathbb{R}$ is a smooth real function, $t\geq0$ 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 $\mathscr{F}^X$ is its natural filtration.