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Kernel estimation for Lévy driven stochastic convolutions

Author

Listed:
  • Comte Fabienne

    (Université de Paris, CNRS, MAP5, UMR 8145, F-75006Paris, France)

  • Genon-Catalot Valentine

    (Université de Paris, CNRS, MAP5, UMR 8145, F-75006Paris, France)

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.

Suggested Citation

  • Comte Fabienne & Genon-Catalot Valentine, 2021. "Kernel estimation for Lévy driven stochastic convolutions," Statistics & Risk Modeling, De Gruyter, vol. 38(1-2), pages 1-24, January.
    • Handle: RePEc:bpj:strimo:v:38:y:2021:i:1-2:p:1-24:n:1
    DOI: 10.1515/strm-2021-0007
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