Kernel estimation for Lévy driven stochastic convolutions

Abstract

Abstract We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X⁢(t)=∫0ta⁢(t-s)⁢dZ⁢(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 (Xi⁢(t))t∈[0,T](X_{i}(t))_{t\in[0,T]}, i=1,…,Ni=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 a2a^{2} under different sets of assumptions. We bound the L2\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.