Consider a moving average process X of the form X(t) = t −∞ ϕ(t − u)dZu, t ≥ 0, where Z is a (non Gaussian) Hermite process of order q ≥ 2 and ϕ : R+ → R is sufficiently integrable. This paper investigates the fluctuations, as T → ∞, of integral functionals of the form t → T t 0 P (X(s))ds, in the case where P is any given polynomial function. It extends a study initiated in [21], where only the quadratic case P (x) = x 2 and the convergence in the sense of finite-dimensional distributions were considered.

We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index H ≥ 1 2 . We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if H > 3 4 and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a "controlled" process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals.

We consider stochastic integrals with respect to standard and fractional Brownian motions. Using Malliavin calculus and fractional integrations techniques, we compute rates of convergence and we prove limit theorems for the approximation of these integrals by their associated Riemann sums.