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

Abdelmalik Keddi, Salim Bouzebda () and Fethi Madani
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Abdelmalik Keddi: Département de Mathématiques et Informatique, Faculté des Sciences et de la Technologie, Université de Tamanghasset, Tamanghasset 11000, Algeria
Salim Bouzebda: LMAC—Laboratory of Applied Mathematics of Compiègne, Université de Technologie de Compiègne, CS 60 319-60 203 Compiègne, France
Fethi Madani: Laboratoire de Modèles Stochastiques, Statistique et Applications, Université Dr. Moulay Tahar de Saida, B. P. 138, En-Nasr, Saida 20000, Algeria

Mathematics, 2025, vol. 13, issue 16, 1-32

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.

Keywords: sub-fractional Brownian motion; trend function; kernel-type estimator; stochastic partial differential equations; non-parametric estimation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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