Nonparametric estimation for i.i.d. Gaussian continuous time moving average models

We consider a Gaussian continuous time moving average model $$X(t)=\int _0^t a(t-s)dW(s)$$ X ( t ) = ∫ 0 t a ( t - s ) d W ( s ) where W is a standard Brownian motion and a(.) a deterministic function locally square integrable on $${{\mathbb {R}}}^+$$ R + . Given N i.i.d. continuous time observations of $$(X_i(t))_{t\in [0,T]}$$ ( X i ( t ) ) t ∈ [ 0 , T ] on [0, T], for $$i=1, \dots , N$$ i = 1 , ⋯ , N distributed like $$(X(t))_{t\in [0,T]}$$ ( X ( t ) ) t ∈ [ 0 , T ] , we propose nonparametric projection estimators of $$a^2$$ a 2 under different sets of assumptions, which authorize or not fractional models. We study the asymptotics in T, N (depending on the setup) ensuring their consistency, provide their nonparametric rates of convergence on functional regularity spaces. Then, we propose a data-driven method corresponding to each setup, for selecting the dimension of the projection space. The findings are illustrated through a simulation study.