Kernel estimation for Lévy driven stochastic convolutions

We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X ⁢ ( t ) = ∫ 0 t a ⁢ ( t - s ) ⁢ d Z ⁢ ( s ) X(t)=\int_{0}^{t}a(t-s)\,dZ(s) , where 𝑍 is a Lévy martingale and the kernel a ( . ) a(\,{.}\,) a deterministic function square integrable on R + \mathbb{R}^{+} . Given 𝑁 i.i.d. continuous time observations ( X i ⁢ ( t ) ) t ∈ [ 0 , T ] (X_{i}(t))_{t\in[0,T]} , i = 1 , … , N i=1,\dots,N , distributed like ( X ⁢ ( t ) ) t ∈ [ 0 , T ] (X(t))_{t\in[0,T]} , we propose two types of nonparametric projection estimators of a 2 a^{2} under different sets of assumptions. We bound the L 2 \mathbb{L}^{2} -risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.