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

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:du(t,x)=(A0+θ(t)A1)u(t,x)dt+dξH(t,x),x∈[0,1],t∈(0,T],u(0,x)=u0(x), where ξH(t,x) is a cylindrical sub-fractional Brownian motion in L2[0,T]×[0,1], and A0+θ(t)A1 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.