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Asymptotic Analysis of a Kernel-Type Estimator for Parabolic Stochastic Partial Differential Equations Driven by Cylindrical Sub-Fractional Brownian Motion

Author

Listed:
  • Abdelmalik Keddi

    (Département de Mathématiques et Informatique, Faculté des Sciences et de la Technologie, Université de Tamanghasset, Tamanghasset 11000, Algeria
    These authors contributed equally to this work.)

  • Salim Bouzebda

    (LMAC—Laboratory of Applied Mathematics of Compiègne, Université de Technologie de Compiègne, CS 60 319-60 203 Compiègne, France
    These authors contributed equally to this work.)

  • Fethi Madani

    (Laboratoire de Modèles Stochastiques, Statistique et Applications, Université Dr. Moulay Tahar de Saida, B. P. 138, En-Nasr, Saida 20000, Algeria)

Abstract

The main purpose of the present paper is to investigate the problem of estimating the time-varying coefficient in a stochastic parabolic equation driven by a sub-fractional Brownian motion. More precisely, we introduce a kernel-type estimator for the time-varying coefficient θ ( t ) in the following evolution equation: d u ( t , x ) = ( A 0 + θ ( t ) A 1 ) u ( t , x ) d t + d ξ H ( t , x ) , x ∈ [ 0 , 1 ] , t ∈ ( 0 , T ] , u ( 0 , x ) = u 0 ( x ) , where ξ H ( t , x ) is a cylindrical sub-fractional Brownian motion in L 2 [ 0 , T ] × [ 0 , 1 ] , and A 0 + θ ( t ) A 1 is a strongly elliptic differential operator. We obtain the asymptotic mean square error and the limiting distribution of the proposed estimator. These results are proved under some standard conditions on the kernel and some mild conditions on the model. Finally, we give an application for the confidence interval construction.

Suggested Citation

  • Abdelmalik Keddi & Salim Bouzebda & Fethi Madani, 2025. "Asymptotic Analysis of a Kernel-Type Estimator for Parabolic Stochastic Partial Differential Equations Driven by Cylindrical Sub-Fractional Brownian Motion," Mathematics, MDPI, vol. 13(16), pages 1-32, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:16:p:2627-:d:1725663
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    References listed on IDEAS

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