Kernel estimation for Lévy driven stochastic convolutions

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